 Regularity for the solutions of linear elliptic equations
 Existence, uniqueness and regularity for evolution equations
Regularity for the solutions of linear elliptic equations
The emphasis in this topic is on the integrability of the gradients of solutions up to an index which is larger than the dimension of the underlying space and, on the other hand, Hölder continuity of the solution under very weak assumptions on the data of the problem. The motivation concerning the integrability property is the insight that exactly this would deliver uniqueness for the solution for many nonlinear models and, additionally, ideas for the approximation of it. The starting point was the result of Gröger, who proved that this can always be achieved in a very general setting if the space dimension equals two. Since then several results have been proved in 3D which partly were already used in several application areas of the institute. The final goal is to establish an appropriate global setting which includes specific discontinuities of the above mentioned type and, nevertheless, assures the required integrability in three dimensions. (It is clear that, firstly, the three dimensional case is the essential one for applications in real space, and, secondly, that the required integrability property cannot be expected in higher dimensions e.g. in case of mixed boundary conditions.) The impetus for the investigation of Hölder continuity came from optimal control, where the corresponding results serve by now as a backbone in the treatment of control problems with pointwise inequality constraints on the state.Existence, uniqueness and regularity for evolution equations
The interest here comes from problems out of semiconductor modeling, elasticity/plasticity/damage modeling, and battery modeling (ion transfer through electrolytes + electrolyte/electrode interaction) at WIAS. Concerning reactiondiffusion equations, an idea of Clement and Li, generalized by Prüss, shows (local in time) existence and uniqueness for these equations, provided one knows maximal parabolic regularity for the  linear  elliptic operator which corresponds to the initial time point and the initial value. Thus, maximal parabolic regularity was proved for general second order divergence operators on different Banach spaces: first on all L^{p} spaces, thus obtaining an instrument for the treatment of equations where the Neumann boundary condition is homogeneous and no distributional right hand sides occur. Second, it was shown together with the Weierstrass awardee Dr. HallerDintelmann that maximal parabolic regularity holds true for elliptic divergence operators in a huge variety of distributional spaces which allow an adequate treatment of the corresponding nonlinear parabolic equations, among them nonlinear heat equations and Van Roosbroeck's system in 3D.Highlights
Globalintime existence analysis for improved NernstPlanckPoisson models
These PDEmodels consist of convectiondiffusionreaction equations describing the masstransfer of chemical species, the equations of momentum balance for the barycentric velocity and the pressure of the mixture (compressible/incompressible NavierStokes equations) and the Poisson equation for the electrical potential. The PDEs in the bulk are supplied with complex boundary conditions describing the surface adsorption of species and their reaction with the components of adjacent materials. The constitutive equations for the diffusion fluxes and the reaction rates are derived by means of modern thermodynamic methods that guarantee consistency with the entropy principle and lead to new PDE structures. Several challenges occur in the analysis of such systems: The Onsager matrix possesses a singular eigenvalue
 Crossdiffusion
 Highly nonlinear reaction rates
 Pressure contribution to the chemical potentials, pressure coupling of the diffusion fluxes
 Compressible NavierStokes system
Existence and regularity of solutions to thermistor systems for organic devices
Organic semiconductor devices are carbonbased electronic devices that show a strong interplay between current and heat flow due to temperatureactivated hopping transport on one hand and Joule selfheating on the other hand. On an empirical level, this electrothermal feedback can be described by means of a p(x)Laplace thermistor model for the electrostatic potential and the temperature, where the p(x)Laplacian takes the nonOhmic behavior of the individual organic layers into account and the temperature dependence is given by an Arrheniuslike law. A mathematical challenge arises from the fact that the Joule selfheating term in the heat equation is a priori only in L^{1}.
Due to missing logHölder continuity of the exponent p(x), known techniques for the treatment of the p(x)Laplacian fail in the setting of organic devices. However, a novel higher integrability result for the gradient of the electrostatic potential was proved in GlitzkyLiero 2017 by means of localization, Caccioppoli estimates, and Gehringtype arguments. The resulting higher integrability of the Joule heat term in the heat flow equation then allows to prove the solvability of the p(x)Laplace thermistor problem by Schauder's fixedpoint theorem. A second approach is presented in BulíčekGlitzkyLiero 2016a. Therein, higher spatial dimensions and arbitrary exponents p(x) are considered. The existence proof is based on a Galerkin approximation of a regularized version of the thermistor system, where the crucial Joule heat term is approximated such that it remains bounded. Finally, in BulíčekGlitzkyLiero 2016b the notion of entropy solutions of the heat equation in conjunction with Schauder's fixed point theorem was investigated. Uniqueness of solutions for such systems cannot be expected due to the hysteretic behavior.
Publications
Monographs

A. Mielke, M. Peletier, D. Slepcev, eds., Variational Methods for Evolution, 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 13911467, (Collection Published), DOI 10.4171/OWR/2020/29 .
Abstract
Variational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also timeincremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations) thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multiagent systems, and in data science. This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, optimal transport, gradient flows, and largedeviation principles for timecontinuous Markov processes, Gammaconvergence and homogenization. 
TH. Eiter, Existence and Spatial Decay of Periodic NavierStokes Flows in Exterior Domains, Th. Eiter, ed., Logos Verlag Berlin GmbH, 2020, 197 pages, (Monograph Published), DOI 10.30819/5108 .
Abstract
A classical problem in the field of mathematical fluid mechanics is the flow of a viscous incompressible fluid past a rigid body. In his doctoral thesis, Thomas Walter Eiter investigates timeperiodic solutions to the associated NavierStokes equations when the body performs a nontrivial translation. The first part of the thesis is concerned with the question of existence of timeperiodic solutions in the case of a nonrotating and of a rotating obstacle. Based on an investigation of the corresponding Oseen linearizations, new existence results in suitable function spaces are established. The second part deals with the study of spatially asymptotic properties of timeperiodic solutions. For this purpose, timeperiodic fundamental solutions to the Stokes and Oseen linearizations are introduced and investigated, and the concept of a timeperiodic fundamental solution for the vorticity field is developed. With these results, new pointwise estimates of the velocity and the vorticity field associated to a timeperiodic fluid flow are derived. 
M. Hintermüller, J.F. Rodrigues, eds., Topics in Applied Analysis and Optimisation  Partial Differential Equations, Stochastic and Numerical Analysis, CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, 396 pages, (Collection Published).

H.Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, xii+571 pages, (Collection Published).
Abstract
This volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; wellposedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a selfcontained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs. 
C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer International Publishing Switzerland, Cham, 2016, xii+155 pages, (Monograph Published).
Abstract
Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an sharmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely selfcontained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance. 
A. Mielke, Chapter 3: On Evolutionary $Gamma$Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 187249, (Chapter Published).
Abstract
In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E_{ε} and the dissipation potential R_{ε} or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions u_{ε}. We investigate the question under which conditions the limits u of (subsequences of) the solutions u_{ε} are solutions of the gradient system generated by the Γlimits E_{0} and R_{0}. Here the choice of the right topology will be crucial as well as additional structural conditions.
We cover classical gradient systems, where R_{ε} is quadratic, and rateindependent systems as well as the passage from viscous to rateindependent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results. 
A. Mielke, T. Roubíček, Rateindependent Systems. Theory and Application, 193 of Applied Mathematical Sciences, Springer International Publishing, New York, 2015, vii+660 pages, (Monograph Published).
Abstract
This monograph provides both an introduction to and a thorough exposition of the theory of rateindependent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rateindependent systems, first on an abstract level either with or even without a linear structure, discussing various concepts of solutions with full mathematical rigor. Then, usefulness of the abstract concepts is demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesivetype contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to ratedependent thermodynamic models are considered as well. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. In this book, the mathematical framework for a rigorous mathematical treatment of "rateindependent systems" is presented in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well written book useful. 
A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125155, (Chapter Published).
Abstract
In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gammaconvergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rateindependent system. On the other hand we show how the concept of BalancedViscosity solution arise as in the vanishingviscosity limit.
As applications we discuss, first, the evolution of laminate microstructures in finitestrain elastoplasticity and, second, a twophase model for shapememory materials, where Hmeasures are used to construct the mutual recovery sequences needed in the existence theory. 
E. Valdinoci, ed., Contemporary PDEs between theory and applications, 35 of Discrete and Continuous Dynamical Systems Series A, American Institute of Mathematical Sciences, Springfield, 2015, 625 pages, (Collection Published).

B. Fiedler, M. Haragus, A. Mielke, G. Raugel, Y. Yi, eds., Special Issue in Memoriam of Klaus Kirchgässner, 27 of J. Dynam. Differential Equations, Springer International Publishing, Cham et al., 2015, 674 pages, (Collection Published).

A. Mielke, Chapter 21: Dissipative Quantum Mechanics Using GENERIC, in: Recent Trends in Dynamical Systems  Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.P. Kruse, F. Rupp, S. Schmitz, eds., 35 of Springer Proceedings in Mathematics & Statistics, Springer, Basel et al., 2013, pp. 555585, (Chapter Published).
Abstract
Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for NonEquilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradientflow contribution, which satisfy a particular noninteraction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a timedependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the KuboMori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative doublebracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finitedimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the FokkerPlanck equation and the entropy gradient flow formulation for reversible Markov chains. 
G. Dal Maso, A. Mielke, U. Stefanelli, eds., Rateindependent Evolutions, 6 (No. 1) of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2013, 275 pages, (Collection Published).

A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., Variational Methods for Evolution, 8 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2011, pp. 31453216, (Chapter Published).

P. Colli, A. Damlamian, N. Kenmochi, M. Mimura, J. Sprekels, eds., Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation, 29 of Gakuto International Series Mathematical Sciences and Applications, Gakkōtosho, Tokyo, 2008, 475 pages, (Collection Published).

U. Bandelow, H. Gajewski, R. Hünlich, Chapter 3: FabryPerot Lasers: Thermodynamicsbased Modeling, in: Optoelectronic Devices  Advanced Simulation and Analysis, J. Piprek, ed., Springer, New York, 2005, pp. 6385, (Chapter Published).

A. Mielke, Chapter 6: Evolution of Rateindependent Systems, in: Handbook of Differential Equations. Vol. 2: Evolutionary Equations, C. Dafermos, E. Feireisl, eds., Elsevier Science B.V., Amsterdam, 2005, pp. 461559, (Chapter Published).

B. Fiedler, K. Gröger, J. Sprekels, eds., EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, 1, World Scientific, Singapore [u. a.], 2000, 806 pages, (Monograph Published).

B. Fiedler, K. Gröger, J. Sprekels, eds., EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, 2, World Scientific, Singapore [u. a.], 2000, 681 pages, (Monograph Published).
Articles in Refereed Journals

P.É. Druet, Globalintime existence for liquid mixtures subject to a generalised incompressibility constraint, Journal of Mathematical Analysis and Applications, 499 (2021), pp. 125059/1125059/56, DOI 10.1016/j.jmaa.2021.125059 .
Abstract
We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a nonsolenoidal velocity field in the NavierStokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDEsystem of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L^{1}. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure. 
A. Stephan, EDPconvergence for a linear reactiondiffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1226/35, DOI 10.1007/s00526021020890 .
Abstract
We perform a fastreaction limit for a linear reactiondiffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reactiondiffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and coshtype functions for the reaction part. The fastreaction limit is done on the level of the gradient structure by proving EDPconvergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slowmanifold. Moreover, the limit gradient system can be equivalently described by a coarsegrained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarsegrained slow variable. 
E. Ipocoana, A. Zafferi, Further regularity and uniqueness results for a nonisothermal CahnHilliard equation, Communications on Pure and Applied Analysis, 20 (2021), pp. 763782, DOI 10.3934/cpaa.2020289 .
Abstract
The aim of this paper is to establish new regularity results for a nonisothermal CahnHilliard system in the twodimensional setting. The main achievement is a crucial L^{∞} estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model. 
D. Bothe, P.É. Druet, Wellposedness analysis of multicomponent incompressible flow models, Journal of Evolution Equations, appeared online on 13.05.2021, DOI 10.1007/s00028021007123 .
Abstract
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the NavierStokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the localintime wellposedness in classes of strong solutions, and the globalintime existence of solutions for initial data sufficiently close to a smooth equilibrium solution. 
P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized CahnHilliard system with fractional operators, Journal of Evolution Equations, 21 (2021), pp. 2749  2778, DOI 10.1007/s00028021007061 .
Abstract
In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous CahnHilliard systems of two operator equations in which nonlinearities of doublewell type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of squareintegrable functions on a bounded and smooth threedimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears. 
P. Colli, A. Signori, J. Sprekels, Secondorder analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. 73/173/46 (Published online on 13.07.2021), DOI 10.1051/cocv/2021072 .
Abstract
This paper concerns a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak wellposedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong wellposedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both firstorder necessary and secondorder sufficient conditions for optimality. The mathematically challenging secondorder analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the secondorder Fréchet derivative of the controltostate operator and carry out a thorough and detailed investigation about the related properties. 
J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. S26/1S26/27, DOI 10.1051/cocv/2020088 .
Abstract
In this paper, we study an optimal control problem for a nonlinear system of reactiondiffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extracellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a doublewell potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L^{1}norm. For such problems, we derive firstorder necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding controltostate operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of secondorder sufficient optimality conditions. 
A.F.M. TER Elst, R. HallerDintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$theory for secondorder elliptic operators in divergence form with complex coefficients, Journal of Evolution Equations, published online on 18.06.2021, DOI 10.1007/s00028021007114 .
Abstract
Given a complex, elliptic coefficient function we investigate for which values of p the corresponding secondorder divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L^{p}(Ω). Additional properties like analyticity of the semigroup, H^{∞}calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of ^{p}'s for small imaginary parts of the coefficients. Our results are based on the recent notion of ^{p}ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients. 
A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal driftdiffusion models with GaussFermi statistics for organic semiconductors, Analysis and Applications, 19 (2021), pp. 275304, DOI 10.1142/S0219530519500246 .
Abstract
This work is concerned with the analysis of a driftdiffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like GaussFermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that selfheating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal driftdiffusion model for organic semiconductors on a sound analytical basis. 
TH. Eiter, On the spatially asymptotic structure of timeperiodic solutions to the NavierStokes equations, Probability Surveys, 149 (2021), pp. 34393451, DOI 10.1090/proc/15482 .
Abstract
The asymptotic behavior of weak timeperiodic solutions to the NavierStokes equations with a drift term in the threedimensional whole space is investigated. The velocity field is decomposed into a timeindependent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions. 
TH. Eiter, G.P. Galdi, Spatial decay of the vorticity field of timeperiodic viscous flow past a body, Archive for Rational Mechanics and Analysis, pp. published online on 06.07.2021 (130), DOI 10.1007/s0020502101690z .
Abstract
We study the asymptotic spatial behavior of the vorticity field associated to a timeperiodic NavierStokes flow past a body in the class of weak solutions satisfying a Serrinlike condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its timeaverage over a period and a socalled purely periodic part, we prove that inside the wake region, the timeaverage has the same algebraic decay as that known for the associated steadystate problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from the body, the timeperiodic vorticity field behaves like the vorticity field of the corresponding steadystate problem. 
TH. Eiter, M. Kyed, Viscous flow around a rigid body performing a timeperiodic motion, Journal of Mathematical Fluid Mechanics, 23 (2021), pp. 28/128/23, DOI 10.1007/s00021021005564 .
Abstract
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed timeperiodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigidbody motion are compatible, and that the mean translational velocity is nonzero, existence of a timeperiodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is illposed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces. 
D. Hömberg, R. Lasarzik, Weak entropy solutions to a model in induction hardening, existence and weakstrong uniqueness, Mathematical Models & Methods in Applied Sciences, 31 (2021), pp. 18671918, DOI 10.1142/S021820252150041X .
Abstract
In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weakstrong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions. 
R. Lasarzik, Analysis of a thermodynamically consistent NavierStokesCahnHilliard model, Nonlinear Analysis. An International Mathematical Journal, 213 (2021), 112526, 33 pp., DOI 10.1016/j.na.2021.112526 .
Abstract
In this paper, existence of generalized solutions to a thermodynamically consistent NavierStokesCahnHilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measurevalued and dissipative solutions. The measurevalued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measurevalued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weakstrong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions. 
X. Liu, E. Titi, Local wellposedness of strong solutions to the threedimensional compressible primitive equations, Archive for Rational Mechanics and Analysis, pp. published online on 26.05.2021 (136), DOI 10.1007/s00205021016623 .
Abstract
This work is devoted to establishing the localintime wellposedness of strong solutions to the threedimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. 
A. Mielke, Relating a rateindependent system and a gradient system for the case of onehomogeneous potentials, Journal of Dynamics and Differential Equations, pp. published online on 31.05.2021 (122), DOI 10.1007/s10884021100073 .
Abstract
We consider a nonnegative and onehomogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradientflow equations and the energetic solutions generated via the rateinpendent system given in terms of the timedependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solutiondependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the totalvariation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constantspeed intervals for the solutins of the gradientflow equation. As a major result we obtain a nontrivial existence and uniqueness result for the energetic rateindependent system. 
A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Journal of Evolution Equations, published online on 19.04.2021, DOI 10.1007/s00028021006862 .
Abstract
We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the waterair interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionallydamped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2. 
W. Dreyer, P.É. Druet, P. Gajewski, C. Guhlke, Analysis of improved NernstPlanckPoisson models of compressible isothermal electrolytes, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 119/1119/68, DOI 10.1007/s00033020013415 .
Abstract
We consider an improved NernstPlanckPoisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the NavierStokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a globalintime weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials. 
P.É. Druet, A. Jüngel, Analysis of crossdiffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 21792197, DOI 10.1137/19M1301473 .
Abstract
The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolichyperbolic system for the mass densities. The globalintime existence of classical and weak solutions is proved in a bounded domain with nopenetration boundary conditions. The idea is to decompose the system into a porousmediumtype equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field. 
N.A. Dao, J.I. Díaz, Q.B.H. Nguyen, Pointwise gradient estimates in multidimensional slow diffusion equations with a singular quenching term, Advanced Nonlinear Studies, 20 (2020), pp. 373384, DOI 10.1515/ans20202076 .

R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 89119 (published online in March 2020), DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
D. Belomestny, J.G.M. Schoenmakers, Optimal stopping of McKeanVlasov diffusions via regression on particle systems, SIAM Journal on Control and Optimization, 58 (2020), pp. 529550, DOI 10.1137/18M1195590 .
Abstract
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKeanVlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered. 
J.A. Carrillo, K. Hopf, M.Th. Wolfram, Numerical study of BoseEinstein condensation in the KaniadakisQuarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507529, DOI 10.3934/krm.2020017 .
Abstract
Kaniadakis and Quarati (1994) proposed a FokkerPlanck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blowup time while having a linear diffusion term. We present a thoroughly validated timeimplicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the KaniadakisQuarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the KaniadakisQuarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blowup profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data. 
J.A. Carrillo, K. Hopf, J.L. Rodrigo, On the singularity formation and relaxation to equilibrium in 1D FokkerPlanck model with superlinear drift, Advances in Mathematics, 360 (2020), pp. 106883/1106883/66, DOI 10.1016/j.aim.2019.106883 .
Abstract
We consider a class of FokkerPlanck equations with linear diffusion and superlineardrift enjoying a formal Wassersteinlike gradient flow structure with convex mobility function. In the driftdominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the onedimensional case, is based on a reformulation of the problem in terms of the pseudoinverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for globalintime existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blowup behaviour, formation of condensates (i.e. Dirac measures at zero) and longtime asymptotics are investigated. As a consequence, in the masssupercritical case,solutions will blow up in L^{∞} in finite time andunderstood in a generalised, measure sensethey will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if d≥3. 
R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$, Journal of Evolution Equations, pp. published online on 20.10.2020, urlhttps://link.springer.com/article/10.1007/s00028020006426, DOI 10.1007/s00028020006426 .
Abstract
We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on L^{p} in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions. 
P. Colli, M.H. Farshbaf Shaker, K. Shirakawa, N. Yamazaki, Optimal control for shape memory alloys of the onedimensional Frémond model, Numerical Functional Analysis and Optimization. An International Journal, 41 (2020), pp. 14211471, DOI 10.1080/01630563.2020.1774892 .
Abstract
In this paper, we consider optimal control problems for the onedimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudopotential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for approximating systems. 
P. Colli, G. Gilardi, J. Sprekels, Asymptotic analysis of a tumor growth model with fractional operators, Asymptotic Analysis, 120 (2020), pp. 4172, DOI 10.3233/ASY191578 .
Abstract
In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of CahnHilliard type modelling tumor growth that has originally been proposed in HawkinsDaarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 324). The original phase field system and certain relaxed versions thereof have been studied in recent papers coauthored by the present authors and E. Rocca. The model consists of a CahnHilliard equation for the tumor cell fraction φ, coupled to a reactiondiffusion equation for a function S representing the nutrientrich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, wellposedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding wellposedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases. 
P. Colli, G. Gilardi, J. Sprekels, Longtime behavior for a generalized CahnHilliard system with fractional operators, Atti della Accademia Peloritana dei Pericolanti. Classe di Scienze, Fisiche, Matematiche e Naturali. AAPP. Physical, Mathematical, and Natural Sciences, 98 (2020), pp. A4/1A4/18, DOI 10.1478/AAPP.98S2A4 .
Abstract
In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the CahnHilliard system, with possibly singular potentials, which we recently investigated in the paper "Wellposedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omegalimit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical potential vanishes at infinity, and every element of the Omegalimit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omegalimit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant. 
TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 395425 (published online in May 2020), DOI 10.3934/dcdss.2020345 .
Abstract
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelinde Donder kinetics. 
J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics, 181 (2020), pp. 22572303, DOI 10.1007/s10955020026634 .
Abstract
We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reactionrate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailedbalance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradientflow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailedbalance steady state. The limit of large volumes is studied in the sense of evolutionary Γconvergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels. 
H. Meinlschmidt, Ch. Meyer, J. Rehberg, Regularization for optimal control problems associated to nonlinear evolution equations, Journal of Convex Analysis, 27 (2020), pp. 443485, DOI 10.20347/WIAS.PREPRINT.2576 .
Abstract
It is wellknown that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard “calculus of variations” proof for the existence of optimal controls. For timedependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochnertype spaces. In this paper, we propose an abstract function space and a suitable regularization or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacletype in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand. 
H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for nonautonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83135, DOI 10.4171/PRIMS/5615 .
Abstract
In the present paper we advocate the HowlandEvans approach to solution of the abstract nonautonomous Cauchy problem (nonACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of Xvalued functions on the timeinterval J. The fundamental observation is a onetoone correspondence between solution operators (propagators) for a nonACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operatortheoretical methods to scrutinise the nonACP including the proof of the Trotter product approximation formulae with operatornorm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces. 
A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for largearea organic lightemitting diodes, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 39533971 (published online on 28.11.2020), DOI 10.3934/dcdss.2020460 .
Abstract
An effective system of partial differential equations describing the heat and current flow through a thin organic lightemitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully threedimensional φ(x)Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the currentflow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the threedimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective currentflow equation is given by two semilinear equations in the twodimensional crosssections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted. 
M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phasefield fracture model at finite strains based on modified invariants, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 100 (2020), pp. e201900288/1e201900288/51, DOI 10.1002/zamm.201900288 .
Abstract
Phasefield models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phasefield fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phasefield model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right CauchyGreen strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the timediscrete solutions converge in a weak sense to a solution of the timecontinuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 95/195/53, DOI 10.1007/s00526020017326 .
Abstract
We study parabolic quasivariational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator. 
P.É. Druet, A theory of generalised solutions for ideal gas mixtures with MaxwellStefan diffusion, Discrete and Continuous Dynamical Systems  Series S, (2020), published online in Nov. 2020, urlhttps://doi.org/10.3934/dcdss.2020458, DOI 10.3934/dcdss.2020458 .
Abstract
After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolichyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the BreschDesjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the MaxwellStefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary MaxwellStefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime. 
A. Mielke, T. Roubíček, Thermoviscoelasticity in KelvinVoigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), pp. 145, DOI 10.1007/s0020502001537z .
Abstract
The frameindifferent thermodynamicallyconsistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the secondgrade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under timedependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization. 
K. Disser, J. Rehberg, The 3D transient semiconductor equations with gradientdependent and interfacial recombination, Mathematical Models & Methods in Applied Sciences, 29 (2019), pp. 18191851, DOI 10.1142/S0218202519500350 .
Abstract
We establish the wellposedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: nonsmooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergenceform operators. 
M. Heida, S. Nesenenko, Stochastic homogenization of ratedependent models of monotone type in plasticity, Asymptotic Analysis, 112 (2019), pp. 185212, DOI 10.3233/ASY181502 .
Abstract
In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for ratedependent systems. The derivations of the homogenization results presented in this work are based on the stochastic twoscale convergence in Sobolev spaces. For completeness, we also present some twoscale homogenization results for convex functionals, which are related to the classical Gammaconvergence theory. 
D. Peschka, S. Haefner, L. Marquant, K. Jacobs, A. Münch, B. Wagner, Signatures of slip in dewetting polymer films, Proceedings of the National Academy of Sciences of the United States of America, 116 (2019), pp. 92759284, DOI 10.1073/pnas.1820487116 .

A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary Gammaconvergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479522, DOI 10.1007/s0002801900484x .
Abstract
We consider the initialvalue problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semiimplicit discretization scheme with a variational approximation technique. 
P. Colli, G. Gilardi, J. Sprekels, A distributed control problem for a fractional tumor growth model, Mathematics  Open Access Journal, 7 (2019), pp. 792/1792/32, DOI 10.3390/math7090792 .
Abstract
In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a CahnHilliard type phase field system modeling tumor growth that goes back to HawkinsDaarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 324.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated controltostate operator, establish the existence of solutions to the associated adjoint system, and derive the firstorder necessary conditions of optimality for a cost functional of tracking type. 
P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general CahnHilliard systems with fractional operators and double obstacle potentials, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 243271 (published online on 21.12.2019), DOI 10.3934/dcdss.2020213 .
Abstract
In the recent paper ”Wellposedness and regularity for a generalized fractional CahnHilliard system”, the same authors derived general wellposedness and regularity results for a rather general system of evolutionary operator equations having the structure of a CahnHilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated doublewell potentials driving the phase separation process modeled by the CahnHilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper ”Optimal distributed control of a generalized fractional CahnHilliard system” (Appl. Math. Optim. (2018), https://doi.org/10.1007/s0024501895407) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic doublewell potentials could be admitted. Results concerning existence of optimizers and firstorder necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated controltostate operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the socalled ”deep quench” method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. FarshbafShaker and J. Sprekels in the paper ”A deep quench approach to the optimal control of an AllenCahn equation with dynamic boundary conditions and double obstacles” (Appl. Math. Optim. 71 (2015), pp. 124) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of CahnHilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate firstorder necessary optimality conditions in terms of a variational inequality and the associated adjoint state system. 
P. Colli, G. Gilardi, J. Sprekels, Recent results on wellposedness and optimal control for a class of generalized fractional CahnHilliard systems, Control and Cybernetics, 48 (2019), pp. 153197.

P. Colli, G. Gilardi, J. Sprekels, Wellposedness and regularity for a fractional tumor growth model, Advances in Mathematical Sciences and Applications, 28 (2019), pp. 343375.
Abstract
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P. Colli, G. Gilardi, J. Sprekels, Wellposedness and regularity for a generalized fractional CahnHilliard system, Rendiconti Lincei  Matematica e Applicazioni, 30 (2019), pp. 437478.
Abstract
In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous CahnHilliard system in which nonlinearities of doublewell type occur. Standard cases like regular or logarithmic potentials, as well as nondifferentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators A and B, where the latter are densely defined, unbounded, selfadjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard secondorder elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourthorder systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general wellposedness and regularity results that extend corresponding results which are known for either the nonfractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. These results are entirely new if at least one of the operators A and B differs from the Laplacian. It turns out that the first eigenvalue λ_{1} of A plays an important und not entirely obvious role: if λ_{1} is positive, then the operators A and B may be completely unrelated; if, however, λ_{1} equals zero, then it must be simple and the corresponding onedimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of B. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system. 
P. Colli, A. Signori, J. Sprekels, Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 83 (2021), pp. 20172049 (published online on 21.10.2019), and 2021 Correction to: Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials (https://doi.org/10.1007/s0024502109771x), DOI 10.1007/s00245019096186 .
Abstract
A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the wellposedness of the state system, the Fréchet differentiability of the controltostate operator in a suitable functional analytic framework, and, lastly, we characterize the firstorder necessary conditions of optimality in terms of a variational inequality involving the adjoint variables. 
P. Dondl, Th. Frenzel, A. Mielke, A gradient system with a wiggly energy and relaxed EDPconvergence, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 68/168/45, DOI 10.1051/cocv/2018058 .
Abstract
If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic system. We introduce a notion of evolutionary Gammaconvergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gammaconvergence. We call this notion relaxed EDPconvergence since the special structure of the dissipation functional may not be preserved under Gammaconvergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential. 
E. Emmrich, R. Lasarzik, Existence of weak solutions to a dynamic model for smecticA liquid crystals under undulations, IMA Journal of Applied Mathematics, 84 (2019), pp. 11431176, DOI 10.1093/imamat/hxz030 .
Abstract
A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smecticA liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the threedimensional case. 
S.P. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels, Strong solutions to nonlocal 2D CahnHilliardNavierStokes systems with nonconstant viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), pp. 678727, DOI 10.1088/13616544/aaedd0 .
Abstract
We consider a nonlinear system which consists of the incompressible NavierStokes equations coupled with a convective nonlocal CahnHilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with noslip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the twodimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weakstrong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal CahnHilliard equation, with a given velocity field, in the three dimensional case as well. 
G. Gilardi, J. Sprekels, Asymptotic limits and optimal control for the CahnHilliard system with convection and dynamic boundary conditions, Nonlinear Analysis. An International Mathematical Journal, 178 (2019), pp. 131, DOI 10.1016/j.na.2018.07.007 .
Abstract
In this paper, we study initialboundary value problems for the CahnHilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous CahnHilliard systems, the `pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the $omega$limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and firstorder necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case. 
V. Laschos, A. Mielke, Geometric properties of cones with applications on the HellingerKantorovich space, and a new distance on the space of probability measures, Journal of Functional Analysis, 276 (2019), pp. 35293576, DOI 10.1016/j.jfa.2018.12.013 .
Abstract
By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the HellingerKantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural twoparameter scaling property of the HellingerKantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a twoparameter rescaling and reparametrization of the geodesics, localangle condition and some partial Ksemiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows. 
J. Sprekels, H. Wu, Optimal distributed control of a CahnHilliardDarcy system with mass sources, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 83 (2021), pp. 489530 (published online on 24.01.2019), DOI 10.1007/s00245019095554 .
Abstract
In this paper, we study an optimal control problem for a twodimensional CahnHilliardDarcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the controltostate operator is Fréchet differentiable between suitable Banach spaces and derive the firstorder necessary optimality conditions in terms of the adjoint variables and the usual variational inequality. 
A.F.M. TER Elst, J. Rehberg, Consistent operator semigroups and their interpolation, Journal of Operator Theory, 82 (2019), pp. 321, DOI 10.7900/jot.2017nov15.2233 .
Abstract
Under a mild regularity condition we prove that the generator of the interpolation of two C_{0}semigroups is the interpolation of the two generators. 
R. Lasarzik, Approximation and optimal control of dissipative solutions to the EricksenLeslie system, Numerical Functional Analysis and Optimization. An International Journal, 40 (2019), pp. 17211767, DOI 10.1080/01630563.2019.1632895 .
Abstract
We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measurevalued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the normrestriction on the director in every step. We introduce a semidiscrete scheme and derive an approximated version of the relativeenergy inequality for solutions of this scheme. Passing to the limit in the semidiscretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semicontinuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity. 
R. Lasarzik, Measurevalued solutions to the EricksenLeslie model equipped with the OseenFrank energy, Nonlinear Analysis. An International Mathematical Journal, 179 (2019), pp. 146183, DOI 10.1016/j.na.2018.08.013 .
Abstract
In this article, we prove the existence of measurevalued solutions to the EricksenLeslie system equipped with the OseenFrank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measurevalued solutions for vanishing regularization. Additionally, it is shown that the measurevalued solution fulfills an energy inequality. 
R. Lasarzik, Weakstrong uniqueness for measurevalued solutions to the EricksenLeslie model equipped with the OseenFrank free energy, Journal of Mathematical Analysis and Applications, 470 (2019), pp. 3690, DOI 10.1016/j.jmaa.2018.09.051 .
Abstract
We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of measurevalued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measurevalued solutions, which fulfill an associated energy inequality, enjoy the weakstrong uniqueness property, i.e. the measurevalued solution agrees with a strong solution if the latter exists. The weakstrong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional. 
M. Liero, S. Melchionna, The weighted energydissipation principle and evolutionary Gammaconvergence for doubly nonlinear problems, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 36/136/38, DOI 10.1051/cocv/2018023 .
Abstract
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the socalled weighted energydissipation (WED) functional, whose minimizer correspond to solutions of an ellipticintime regularization of the target problems with regularization parameter δ. We investigate the relation between the Γconvergence of the WED functionals and evolutionary Γconvergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γconvergence of functionals or in the sense of evolutionary Γconvergence of functionaldriven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λconvex energy functionals. Finally, we discuss a homogenization problem as an example of application. 
A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligandreceptor bulksurface system on a moving domain: Well posedness, regularity and convergence to equilibrium, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 15441592, DOI 10.1137/16M110808X .
Abstract
We prove existence, uniqueness, and regularity for a reactiondiffusion system of coupled bulksurface equations on a moving domain modelling receptorligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the nonmoving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgitype arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for timedependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves. 
M. Heida, M. Röger, Large deviation principle for a stochastic AllenCahn equation, Journal of Theoretical Probability, 31 (2018), pp. 364401, DOI 10.1007/s1095901607117 .
Abstract
The AllenCahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reactiondiffusion equation. Stochastic perturbations, especially in the case of additive noise, to the AllenCahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber [Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013)]. We first provide a large deviation principle for stochastic flows in spaces of functions that are Höldercontinuous in time, which extends results by Budhiraja, Dupuis and Maroulas [Ann. Probab. 36 (2008)]. From this result and a continuity argument we deduce a large deviation principle for the AllenCahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional. 
M. Heida, On convergences of the squareroot approximation scheme to the FokkerPlanck operator, Mathematical Models & Methods in Applied Sciences, 28 (2018), pp. 25992635, DOI 10.1142/S0218202518500562 .
Abstract
We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the FokkerPlanck equation using a discrete notion of Gconvergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the FokkerPlanck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic twoscale convergence to prove that this setting satisfies the Gconvergence property. In particular, the class of tessellations for which the Gconvergence result holds is not trivial. 
E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 97 (2018), pp. 012801/1012801/12, DOI 10.1103/PhysRevE.97.012801 .
Abstract
We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the timedependent and nonuniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the MullinsSekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharpinterface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries. 
A. Ceretani, C.N. Rautenberg, The Boussinesq system with mixed nonsmooth boundary conditions and ``donothing'' boundary flow, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 14/114/24 (published online on 07.12.2018), DOI 10.1007/s000330181058y .
Abstract
A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel nonsmooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the donothing boundary condition for the fluid: 1) the directional donothing condition and 2) the donothing condition together with an integral bound for the backflow. Wellposedness of variational formulations is proved for each problem. 
K. Disser, M. Liero, J. Zinsl, On the evolutionary Gammaconvergence of gradient systems modeling slow and fast chemical reactions, Nonlinearity, 31 (2018), pp. 36893706, DOI 10.1088/13616544/aac353 .
Abstract
We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of massaction type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an Econvergence result via Γconvergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudometric. 
M. Becker, Th. Frenzel, Th. Niedermeyer, S. Reichelt, A. Mielke, M. Bär, Local control of globally competing patterns in coupled SwiftHohenberg equations, Chaos. An Interdisciplinary Journal of Nonlinear Science, 28 (2018), pp. 043121/1043121/11, DOI 10.1063/1.5018139 .
Abstract
We present analytical and numerical investigations of two antisymmetrically coupled 1D SwiftHohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimensiontwo point of the Turingwave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left and right traveling waves. In particular, these complex GinzburgLandautype equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other; and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results. 
D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKeanVlasov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 31693195, DOI 10.1137/17M1111024 .
Abstract
We propose a novel projectionbased particle method for solving McKeanVlasov stochastic differential equations. Our approach is based on a projectiontype estimation of the marginal density of the solution in each time step. The projectionbased particle method leads in many situations to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis, particularly in the case of linearly growing coefficients, turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKeanVlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples. 
P. Colli, G. Gilardi, J. Sprekels, On a CahnHilliard system with convection and dynamic boundary conditions, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 197 (2018), pp. 14451475, DOI 10.1007/s1023101807321 .
Abstract
This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of CahnHilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure CahnHilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a FaedoGalerkin scheme, is introduced and rigorously discussed. 
P. Colli, G. Gilardi, J. Sprekels, On the longtime behavior of a viscous CahnHilliard system with convection and dynamic boundary conditions, Journal of Elliptic and Parabolic Equations, 4 (2018), pp. 327347, DOI 10.1007/s4180801800216 .
Abstract
In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a threedimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective CahnHilliard system, which consists of two nonlinearly coupled secondorder partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energies functionals that may be nondifferentiable and have derivatives only in the sense of (possibly setvalued) subdifferentials. For the resulting initialboundary value system of CahnHilliard type, general wellposedness results have been established in piera recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the ωlimit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the omegalimit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant. 
P. Colli, G. Gilardi, J. Sprekels, Optimal distributed control of a generalized fractional CahnHilliard system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 82 (2020), pp. 551589 (published online on 15.11.2018), DOI 10.1007/s0024501895407 .
Abstract
In the recent paper “Wellposedness and regularity for a generalized fractional CahnHilliard system” by the same authors, general wellposedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous CahnHilliard system, in which nonlinearities of doublewell type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B, having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated controltostate mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and firstorder necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system. 
E. Emmrich, S.H.L. Klapp, R. Lasarzik, Nonstationary models for liquid crystals: A fresh mathematical perspective, Journal of NonNewtonian Fluid Mechanics, 259 (2018), pp. 3247, DOI 10.1016/j.jnnfm.2018.05.003 .

E. Emmrich, R. Lasarzik, Existence of weak solutions to the EricksenLeslie model for a general class of free energies, Mathematical Methods in the Applied Sciences, 41 (2018), pp. 64926518.

E. Emmrich, R. Lasarzik, Weakstrong uniqueness for the general EricksenLeslie system in three dimensions, Discrete and Continuous Dynamical Systems, 38 (2018), pp. 46174635, DOI 10.3934/dcds.2018202 .

P. Gurevich, S. Reichelt, Pulses in FitzHughNagumo systems with rapidly oscillating coefficients, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 833856.
Abstract
This paper is devoted to pulse solutions in FitzHughNagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a twoscale FitzHughNagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulselike solutions of the original system. 
J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energyreactiondiffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 10371075, DOI 10.1137/16M1062065 .
Abstract
We derive thermodynamically consistent models of reactiondiffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusionreaction bipolar energy transport system, and a driftdiffusionreaction energy transport system with confining potential. We prove corresponding entropyentropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L^{1} using CziszarKullbackPinsker type inequalities. 
G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rateindependent damage in thermoviscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 13111364, DOI 10.1007/s108840189666y .
Abstract
We present a model for rateindependent, unidirectional, partial damage in viscoelastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rateindependent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the AmbrosioTortorelli phasefield model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled timediscrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rateindependent limit model for displacements and damage, which is independent of temperature. 
A. Muntean, S. Reichelt, Corrector estimates for a thermodiffusion model with weak thermal coupling, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 807832, DOI 10.1137/16M109538X .
Abstract
The present work deals with the derivation of corrector estimates for the twoscale homogenization of a thermodiffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged highcontrast microstructures. The terminology “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conductiondiffusion interaction terms, while the “highcontrast” is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the firstorder terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufourlike effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with εindependent estimates for the thermal and concentration fields and for their coupled fluxes 
M. Thomas, C. Bilgen, K. Weinberg, Phasefield fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMMMitteilungen, 40 (2018), pp. 207237, DOI 10.1002/gamm.201730004 .
Abstract
Phasefield models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phasefield fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phasefield model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right CauchyGreen strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions %Second the mathematical background of the approach is examined and and we show that the timediscrete solutions converge in a weak sense to a solution of the timecontinuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results. 
R. Lasarzik, Dissipative solution to the EricksenLeslie system equipped with the OseenFrank energy, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 39 (published online on 29.11.2018, urlhttps://doi.org/10.1007/s0003301810533, DOI 10.1007/s0003301810533 .
Abstract
We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measurevalued solutions to the considered system and showed global existence as well as weakstrong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weakstrong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the longtime behavior and show that all solutions converge for the large time limit to a certain steady state. 
M. Liero, S. Melchionna, The weighted energydissipation principle and evolutionary Gammaconvergence for doubly nonlinear problems, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 36/136/38, DOI 10.1051/cocv/2018023 .
Abstract
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the socalled weighted energydissipation (WED) functional, whose minimizer correspond to solutions of an ellipticintime regularization of the target problems with regularization parameter δ. We investigate the relation between the Γconvergence of the WED functionals and evolutionary Γconvergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γconvergence of functionals or in the sense of evolutionary Γconvergence of functionaldriven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λconvex energy functionals. Finally, we discuss a homogenization problem as an example of application. 
M. Liero, S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary Gammaconvergence, NoDEA. Nonlinear Differential Equations and Applications, 25 (2018), pp. 6/16/31, DOI 10.1007/s0003001804959 .
Abstract
In this paper we discuss two approaches to evolutionary Γconvergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γconvergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the timedependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energydissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for CahnHilliardtype equations. Using the method of weak and strong twoscale convergence via periodic unfolding, we show that the energy and dissipation functionals Γconverge. In conclusion, we will give specific examples for the applicability of each of the two approaches. 
S. Bergmann, D.A. BarraganYani, E. Flegel, K. Albe, B. Wagner, Anisotropic solidliquid interface kinetics in silicon: An atomistically informed phasefield model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1065015/20, DOI 10.1088/1361651X/aa7862 .
Abstract
We present an atomistically informed parametrization of a phasefield model for describing the anisotropic mobility of liquidsolid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the StillingerWeber interatomic potential. The temperaturedependent interface velocity follows a VogelFulcher type behavior and allows to properly account for the dynamics in the undercooled melt. 
K. Disser, A.F.M. TER Elst, J. Rehberg, On maximal parabolic regularity for nonautonomous parabolic operators, Journal of Differential Equations, 262 (2017), pp. 20392072.
Abstract
We consider linear inhomogeneous nonautonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic L^{r}regularity for discontinuous nonautonomous secondorder divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations. 
K. Disser, J. Rehberg, A.F.M. TER Elst, Hölder estimates for parabolic operators on domains with rough boundary, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XVII (2017), pp. 6579.
Abstract
In this paper we investigate linear parabolic, secondorder boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain  including a very weak compatibility condition between the Dirichlet boundary part and its complement  we prove Hölder continuity of the solution in space and time. 
S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Asymptotic Analysis, 105 (2017), pp. 326, DOI 10.3233/ASY171432 .
Abstract
Based on previous homogenization results for imperfect transmission problems in twocomponent domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , ½] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method. 
M. Heida, A. Mielke, Averaging of timeperiodic dissipation potentials in rateindependent processes, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 13031327.
Abstract
We study the existence and wellposedness of rateindependent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equicontinuity of the solutions in the limit ε→0. 
M. Heida, Stochastic homogenization of rateindependent systems, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 853894, DOI 10.1007/s001610170564z .
Abstract
We study the stochastic and periodic homogenization 1homogeneous convex functionals. We proof some convergence results with respect to stochastic twoscale convergence, which are related to classical Gammaconvergence results. The main result is a general liminfestimate for a sequence of 1homogeneous functionals and a twoscale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of PrandltReuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure. 
M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 135, DOI 10.3934/dcdss.2017001 .
Abstract
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general nonquadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of largedeviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to coshtype dissipation potentials. A second origin arises via a new form of convergence, that we call EDPconvergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gammalimit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reactiondiffusion system. 
CH. Dörlemann, M. Heida, B. Schweizer, Transmission conditions for the Helmholtz equation in perforated domains, Vietnam Journal of Mathematics, 45 (2017), pp. 241253, DOI 10.1007/s100130160222y .

R. Rossi, M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 14191494.
Abstract
We address the analysis of an abstract system coupling a rateindependet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised timediscretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rateindependent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rateindependent processes in viscoelastic solids with inertia, and to a recently proposed model for damage with plasticity. 
R. Rossi, M. Thomas, From adhesive to brittle delamination in viscoelastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 14891546, DOI 10.1142/S0218202517500257 .
Abstract
In this paper we analyze a system for brittle delamination between two viscoelastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rateindependent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rateindependent systems to the present mixed ratedependent/rateindependent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 25182546.
Abstract
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by HawkinsDaruud et al. in citeHZO. The model consists of a CahnHilliard equation for the tumor cell fraction $vp$ coupled to a reactiondiffusion equation for a function $s$ representing the nutrientrich extracellular water volume fraction. The distributed control $u$ monitors as a righthand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the controltostate operator is Fréchet differentiable between appropriate Banach spaces and derive the firstorder necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a CahnHilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 3754.
Abstract
This paper is concerned with a phase field system of CahnHilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of wellposedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates 
P. Colli, G. Gilardi, J. Sprekels, Global existence for a nonstandard viscous CahnHilliard system with dynamic boundary condition, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 17321760, DOI 10.1137/16M1087539 .
Abstract
In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by PodioGuidugli in Ric. Mat. 55 (2006), pp. 105118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the LaplaceBeltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different wellposedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies. 
CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperaturedependent phasefield model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177247.
Abstract
In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for CahnHilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461490] in the framework of FourierNavierStokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 13451369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 25192586] for the study of PDE systems for phase transition and damage. Our globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme. 
P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with thermal phase transition, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 26752710, DOI 10.1142/S0218202517500555 .
Abstract
In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquidsolid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the ClausiusDuhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cutoff techniques and suitable Mosertype estimates. 
H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 1: Existence of optimal solutions, SIAM Journal on Control and Optimization, 55 (2017), pp. 28762904, DOI 10.1137/16M1072644 .
Abstract
This paper is concerned with the stateconstrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive firstorder necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results. 
H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 2: Optimality conditions, SIAM Journal on Control and Optimization, 55 (2017), pp. 23682392, DOI 10.1137/16M1072656 .
Abstract
This paper is concerned with the stateconstrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive firstorder necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results. 
A. Roggensack, Ch. Kraus, Existence of weak solutions for the CahnHilliard reaction model including elastic effects and damage, Journal of Partial Differential Equations, 30 (2017), pp. 111145, DOI 10.4208/jpde.v30.n2.2 .
Abstract
In this paper, we introduce and study analytically a vectorial CahnHilliard reaction model coupled with ratedependent damage processes. The recently proposed CahnHilliard reaction model can e.g. be used to describe the behavior of electrodes of lithiumion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithiumion batteries. Mathematically, this is realized by a CahnLarché system with a nonlinear Newton boundary condition for the chemical potential and a doubly nonlinear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the nonlinear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system. 
J. Sprekels, E. Valdinoci, A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM Journal on Control and Optimization, 55 (2017), pp. 7093.
Abstract
In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. These results are then employed to derive existence as well as firstorder necessary and secondorder sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the “control parameter” that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new classof identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter also the domain of definition, and thus the underlying function space, of the fractional operator changes. 
A. Glitzky, M. Liero, Analysis of p(x)Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 34 (2017), pp. 536562.
Abstract
We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring nonOhmic currentvoltage laws and selfheating effects. The coupled system consists of the currentflow equation for the electrostatic potential and the heat equation with Joule heating term as source. The selfheating in the device is modeled by an Arrheniuslike temperature dependency of the electrical conductivity. Moreover, the nonOhmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$Laplacetype problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the twodimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppolitype estimates, Poincaré inequalities, and a Gehringtype Lemma for the $p(x)$Laplacian. Finally, Schauder's fixedpoint theorem is used to show the existence of solutions. 
M. Thomas, Ch. Zanini, Cohesive zonetype delamination in viscoelasticity, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 14871517, DOI 10.3934/dcdss.2017077 .
Abstract
We study a model for the rateindependent evolution of cohesive zone delamination in a viscoelastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.
Due to the presence of multivalued and unbounded operators featuring nonpenetration and the `memory'constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIASPreprint2113]. 
N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, A review of variational multiscale methods for the simulation of turbulent incompressible flows, Archives of Computational Methods in Engineering. State of the Art Reviews, 24 (2017), pp. 115164.
Abstract
Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible NavierStokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized. 
N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, Analysis of a full spacetime discretization of the NavierStokes equations by a local projection stabilization method, IMA Journal of Numerical Analysis, 37 (2017), pp. 14371467, DOI 10.1093/imanum/drw048 .
Abstract
A finite element error analysis of a local projection stabilization (LPS) method for the timedependent NavierStokes equations is presented. The focus is on the highorder termbyterm stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projectionstabilized structure of standard LPS methods is replaced by an interpolationstabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy. 
N. Ahmed, On the graddiv stabilization for the steady Oseen and NavierStokes equations, Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 54 (2017), pp. 471501, DOI 10.1007/s100920160194z .
Abstract
This paper studies the parameter choice in the graddiv stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H^{1}(Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H^{1}(Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier Stokes equations, numerical simulations were performed on a twodimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without graddiv stabilization. 
M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasivariational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 135.
Abstract
A class of abstract nonlinear evolution quasivariational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semidiscrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradienttype. 
A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energyreactiondiffusion systems using vectorvalued functional inequalities, Journal of Nonlinear Science, 28 (2018), pp. 765806 (published online on 11.11.2017), DOI 10.1007/s0033201794279 .
Abstract
We discuss how the recently developed energydissipation methods for reactiondi usion systems can be generalized to the nonisothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the logSobolev estimate and variants for lowerorder entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method. 
A. Mielke, C. Patz, Uniform asymptotic expansions for the infinite harmonic chain, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 36 (2017), pp. 437475, DOI 10.4171/ZAA/1596 .
Abstract
We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function. 
A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Nonequilibrium thermodynamical principles for chemical reactions with massaction kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 15621585, DOI 10.1137/16M1102240 .
Abstract
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a nonlinear relation between thermodynamic fluxes and free energy driving force. 
E. Cinti, J. Davila, M. Del Pino, Solutions of the fractional AllenCahn equation which are invariant under screw motion, Journal of the London Mathematical Society. Second Series, 94 (2016), pp. 295313.
Abstract
We establish existence and nonexistence results for entire solutions to the fractional AllenCahn equation in R3 , which vanish on helicoids and are invariant under screwmotion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature. 
K. Disser, G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid, Archive for Rational Mechanics and Analysis, 221 (2016), pp. 487526.
Abstract
We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (NavierStokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result.
In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases. 
A. Alphonse, Ch.M. Elliott, Wellposedness of a fractional porous medium equation on an evolving surface, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016), pp. 342.

M. Cozzi, A. Farina, E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Advances in Mathematics, 293 (2016), pp. 343381.
Abstract
We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classi?cation results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of nonsingular and non degenerate anisotropic equations. 
S.P. Frigeri, Global existence of weak solutions for a nonlocal model for twophase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 19571993.
Abstract
We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a NavierStokes type system coupled with a convective nonlocal CahnHilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular doublewell potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model. 
M. Heida, B. Schweizer, Nonperiodic homogenization of infinitesimal strain plasticity equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016), pp. 523.

M. Liero, A. Mielke, G. Savaré, Optimal transport in competition with reaction: The HellingerKantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 28692911.
Abstract
We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ ^{d}, which we call HellingerKantorovich distance. It can be seen as an infconvolution of the wellknown KantorovichWasserstein distance and the HellingerKakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties. 
D. Peschka, N. Rotundo, M. Thomas, Towards doping optimization of semiconductor lasers, Journal of Computational and Theoretical Transport, 45 (2016), pp. 410423.
Abstract
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantumwell lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights. 
M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016), pp. 189210.
Abstract
We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible NavierStokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 . 
S. Dipierro, O. Savin, E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calculus of Variations and Partial Differential Equations, 55 (2016), pp. 86/186/25.
Abstract
In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler?Lagrange equation related to the nonlocal mean curvature. 
S. Patrizi, E. Valdinoci, Relaxation times for atom dislocations in crystals, Calculus of Variations and Partial Differential Equations, 55 (2016), pp. 71/171/44.
Abstract
We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls?Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. In this sense, the system exhibits two different decay behaviors, namely an exponential time decay versus a polynomial decay in the space variables (and these two homogeneities are kept separate during the time evolution). 
P. Bringmann, C. Carstensen, Ch. Merdon, Guaranteed error control for the pseudostress approximation of the Stokes equations, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 14111432.
Abstract
The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the RaviartThomas discretization which is related to the CrouzeixRaviart nonconforming finite element scheme in the lowestorder case. The effective and guaranteed a posteriori error control for this nonconforming velocityoriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local infsup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy. 
M. Bulíček, A. Glitzky, M. Liero, Systems describing electrothermal effects with p(x)Laplacian like structure for discontinuous variable exponents, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 34963514.
Abstract
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L^{1} term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the nonOhmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles  the discontinuous variable exponent (which limits the use of standard methods) and the L^{1} right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes. 
P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous CahnHilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016), pp. 195225, DOI 10.1007/s002450159299z .
Abstract
A boundary control problem for the viscous CahnHilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved. 
P. Colli, G. Gilardi, J. Sprekels, Constrained evolution for a quasilinear parabolic equation, Journal of Optimization Theory and Applications, 170 (2016), pp. 713734.
Abstract
In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the wellposedness and some regularity results for the CauchyNeumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L^{2}(Ω). Then, we consider convex sets of obstacle or doubleobstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property. 
P. Colli, G. Gilardi, J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), pp. 246281.
Abstract
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous CahnHilliard type. The model constitutes a nonlocal version of a model for twospecies phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. PodioGuidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the firstorder necessary conditions of optimality. 
P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal CahnHilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016), pp. 79407964.
Abstract
This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. PodioGuidugli in Ric. Mat. 55 (2006) 105118. The model consists of an initialboundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or doubleobstacle potentials are admitted. In contrast to the local model, which was studied by P. PodioGuidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible longrange interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space. 
J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016), pp. 357380.
Abstract
We construct codimension 11 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts). 
A. Farina, E. Valdinoci, 1D symmetry for semilinear PDEs from the limit interface of the solution, Communications in Partial Differential Equations, 41 (2016), pp. 665682.
Abstract
We study bounded, monotone solutions of ?u = W?(u) in the whole of ?n, where W is a doublewell potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of ?convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4. 
R. HallerDintelmann, A. Jonsson, D. Knees, J. Rehberg, Elliptic and parabolic regularity for second order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016), pp. 50075026, DOI 10.1002/mma.3484/abstract .
Abstract
We study second order equations and systems on nonLipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. 
H. Meinlschmidt, J. Rehberg, Hölderestimates for nonautonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016), pp. 147184.
Abstract
In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on nonsmooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations. 
X. RosOton, E. Valdinoci, The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains, Advances in Mathematics, 288 (2016), pp. 732790.
Abstract
We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of onedimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample. 
S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Advances in Mathematical Sciences and Applications, 25 (2016), pp. 117131.
Abstract
This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly εperiodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and nondegenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale. 
N. Ahmed, G. Matthies, Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to timedependent convectiondiffusionreaction equations, Journal of Scientific Computing, 67 (2016), pp. 9881018.
Abstract
This paper considers the numerical solution of timedependent convectiondiffusionreaction equations. We shall employ combinations of streamlineupwind PetrovGalerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous GalerkinPetrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the longtime behavior of overshoots and undershoots is investigated. 
M.H. Farshbaf Shaker, C. Hecht, Optimal control of elastic vectorvalued AllenCahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016), pp. 129152.
Abstract
In this paper we consider a elastic vectorvalued AllenCahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized firstorder necessary optimality conditions for the original problem. 
S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal CahnHilliard/NavierStokes system in two dimensions, SIAM Journal on Control and Optimization, 54 (2016), pp. 221  250.
Abstract
We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the NavierStokes system with a convective nonlocal CahnHilliard equation in two dimensions of space. We apply recently proved wellposedness and regularity results in order to establish existence of optimal controls as well as firstorder necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow. 
CH. Heinemann, K. Sturm, Shape optimisation for a class of semilinear variational inequalities with applications to damage models, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 35793617, DOI 10.1137/16M1057759 .
Abstract
The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish stateshape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for timediscretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields. 
A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of NonEquilibrium Thermodynamics, 41 (2016), pp. 141149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
A. Mielke, T. Roubíček, Rateindependent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 22032236.
Abstract
Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rateindependent evolution. The energy functional with a frameindifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonselfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.
In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations. 
A. Mielke, R. Rossi, G. Savaré, Balanced viscosity (BV) solutions to infinitedimensional rateindependent systems, Journal of the European Mathematical Society (JEMS), 18 (2016), pp. 21072165.
Abstract
Balanced Viscosity solutions to rateindependent systems arise as limits of regularized rateindependent ows by adding a superlinear vanishingviscosity dissipation. We address the main issue of proving the existence of such limits for innitedimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energydissipation identity. A careful description of the jump behavior of the solutions, of their dierentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chainrule inequality for functions of bounded variation in Banach spaces, on rened lower semicontinuitycompactness arguments, and on new BVestimates that are of independent interest. 
R.I.A. Patterson, Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries, Journal of Evolution Equations, 16 (2016), pp. 261291.
Abstract
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semilinear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semigroups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit. 
K. Disser, M. Meyries, J. Rehberg, A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces, Journal of Mathematical Analysis and Applications, 430 (2015), pp. 11021123.
Abstract
In this paper we consider scalar parabolic equations in a general nonsmooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem. 
K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015), pp. 233253.
Abstract
We study the approximation of Wasserstein gradient structures by their finitedimensional analog. We show that simple finitevolume discretizations of the linear FokkerPlanck equation exhibit the recently established entropic gradientflow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradientflow structures. In particular, we make no use of the linearity of the equations nor of the fact that the FokkerPlanck equation is of second order. 
K. Disser, H.Chr. Kaiser, J. Rehberg, Optimal Sobolev regularity for linear secondorder divergence elliptic operators occurring in realworld problems, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 17191746.
Abstract
On bounded threedimensional domains, we consider divergencetype operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from realworld applications are provided in great detail. 
K. Disser, Wellposedness for coupled bulkinterface diffusion with mixed boundary conditions, Analysis. International Mathematical Journal of Analysis and its Applications, 35 (2015), pp. 309317.

S. Patrizi, E. Valdinoci, Crystal dislocations with different orientations and collisions, Archive for Rational Mechanics and Analysis, 217 (2015), pp. 231261.
Abstract
We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of PeierlsNabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. 
S. Patrizi, E. Valdinoci, Homogenization and Orowan's law for anisotropic fractional operators of any order, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 119 (2015), pp. 336.
Abstract
We consider an anisotropic fractional operator and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the fractional parameter. In the isotropic onedimensional case, we also prove a statement related to the socalled Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior. 
E. Rocca, R. Rossi, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM Journal on Mathematical Analysis, 74 (2015), pp. 25192586.
Abstract
In this paper we analyze a PDE system modelling (nonisothermal) phase transitions and dam age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the righthand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as “entropic”, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain globalintime existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its “entropic” formulation), and of the a priori estimates performed on it. Our timediscrete analysis could be useful towards the numerical study of this model. 
E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective CahnHilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015), pp. 16541680.
Abstract
In this paper we study a distributed optimal control problem for a nonlocal convective CahnHilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the wellposedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate firstorder necessary optimality conditions for the optimal control problem. 
P.É. Druet, Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some lowfrequency Maxwell equations, Discrete and Continuous Dynamical Systems, 8 (2015), pp. 479496.
Abstract
We show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A DivCurl Lemmatype argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the lowfrequency approximation of the Maxwell equations in timeharmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities. 
P.É. Druet, Some mathematical problems related to the second order optimal shape of a crystallization interface, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 24432463.
Abstract
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solidliquid interface in a twophase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient. 
S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for twophase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), pp. 12571293.
Abstract
We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the NavierStokes system coupled with a convective nonlocal CahnHilliard equation with nonconstant mobility. We first prove the existence of a global weak solution in the case of nondegenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal CahnHilliard equation with degenerate mobility and singular potential in dimension three. 
M. Heida, Existence of solutions for two types of generalized versions of the CahnHilliard equation, Applications of Mathematics, 60 (2015), pp. 5190.
Abstract
We show existence of solutions to two types of generalized anisotropic CahnHilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical noflux boundary conditions where the mobility depends on concentration u, gradient of concentration ?u and the chemical potential ?u?s?(u). The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures. 
M. Heida, On systems of CahnHilliard and AllenCahn equations considered as gradient flows in Hilbert spaces, Journal of Mathematical Analysis and Applications, 423 (2015), pp. 410455.

M. Liero, Th. Koprucki, A. Fischer, R. Scholz, A. Glitzky, pLaplace thermistor modeling of electrothermal feedback in organic semiconductors, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 29572977.
Abstract
In largearea Organic LightEmitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these selfheating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a pLaplacetype equation for the electrostatic potential with mixed boundary conditions. The pLaplacian describes the nonOhmic electrical behavior of the organic material. Moreover, an Arrheniuslike temperature dependency of the electrical conductivity is considered. We introduce a finitevolume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem. 
S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delaydifferential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 537553.
Abstract
The subject of the paper are scalar delaydifferential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delaydifferential equations close to the destabilization threshold. 
S. Yanchuk, G. Giacomelli, Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatiotemporal equivalence, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 92 (2015), pp. 042903/1042903/12.
Abstract
Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a StuartLandau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a spacetime representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the spacetime representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms. 
E. Bonetti, Ch. Heinemann, Ch. Kraus, A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, Journal of Partial Differential Equations, 258 (2015), pp. 39283959.
Abstract
In this work, we analytically investigate a multicomponent system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques. 
A. Di Castro, M. Novaga, R. Berardo, E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calculus of Variations and Partial Differential Equations, 54 (2015), pp. 24212464.

S. Dipierro, E. Valdinoci, A density property for fractional weighted Sobolev spaces, Rendiconti Lincei  Matematica e Applicazioni, 26 (2015), pp. 397422.
Abstract
In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant. 
S. Dipierro, E. Valdinoci, On a fractional harmonic replacement, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 33773392.
Abstract
Given $s ∈(0,1)$, we consider the problem of minimizing the Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions. 
S. Dipierro, O. Savin, E. Valdinoci, A nonlocal free boundary problem, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 45594605.
Abstract
We consider a nonlocal free boundary problem built by a fractional Dirichlet norm plus a fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper. 
R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermoviscoelasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015), pp. 159.
Abstract
We address the analysis of a model for brittle delamination of two viscoelastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques. 
R. Servadei, E. Valdinoci, The BrezisNirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367 (2015), pp. 67102.

P. Auscher, N. Badr, R. HallerDintelmann, J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary condition on $L^p$, Journal of Evolution Equations, 15 (2015), pp. 165208.

P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a CahnHilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015), pp. 93108.
Abstract
In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of CahnHilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [ColliGilardiHilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system. 
P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the pure CahnHilliard equation with dynamic boundary conditions, Advances in Nonlinear Analysis, 4 (2015), pp. 311325.
Abstract
A boundary control problem for the pure CahnHilliard equations with possibly singular potentials and dynamic boundary conditions is studied and firstorder necessary conditions for optimality are proved. 
P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous CahnHilliard system with dynamic boundary condition and double obstacle potentials, SIAM Journal on Control and Optimization, 53 (2015), pp. 26962721.
Abstract
In this paper, we investigate optimal boundary control problems for CahnHilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the LaplaceBeltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive firstorder necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, FarshbafShaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) AllenCahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a socalled “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Secondorder analysis of a boundary control problem for the viscous CahnHilliard equation with dynamic boundary conditions, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 7 (2015), pp. 4166.
Abstract
In this paper we establish secondorder sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous CahnHilliard equation with possibly singular potentials and dynamic boundary conditions. 
P. Colli, M.H. Farshbaf Shaker, J. Sprekels, A deep quench approach to the optimal control of an AllenCahn equation with dynamic boundary conditions and double obstacles, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 71 (2015), pp. 124.
Abstract
In this paper, we investigate optimal control problems for AllenCahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the LaplaceBeltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive firstorder necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a socalled “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Analysis & PDE, 8 (2015), pp. 11651235.
Abstract
For a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function in the expanding domain. 
M. Egert, R. HallerDintelmann, J. Rehberg, Hardy's inequality for functions vanishing on a part of the boundary, Potential Analysis, 43 (2015), pp. 4978.
Abstract
We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary. 
M.M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), pp. 19371961.
Abstract
We consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique. 
E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the BallMajumdar free energy, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015), pp. 12691299.
Abstract
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landaude Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible NavierStokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Qtensor $QQ$, namely a tensorvalued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Qtensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blowup rate, is carried out. 
A. Fiscella, R. Servadei, E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 35513563.
Abstract
In this paper we study a nonlocal fractional Laplace equation, depending on a parameter, with asymptotically linear righthand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one. 
D.A. Gomes, S. Patrizi, Obstacle meanfield game problem, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 17 (2015), pp. 5568.
Abstract
In this paper, we introduce and study a firstorder meanfield game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and powerlike nonlinearities. Since the obstacle operator is not differentiable, the equations for firstorder mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions. 
F. Punzo, E. Valdinoci, Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations, Journal of Differential Equations, 258 (2015), pp. 555587.

T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stressdriven localsolution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 645663.
Abstract
A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rateindependent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semiimplicit scheme and regularized by a BVtype gradient term. An analytical zerodimensional example motivates the model and a specific localsolution concept. Twodimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli. 
A.F.M. TER Elst, J. Rehberg, Hölder estimates for secondorder operators with mixed boundary conditions, Advances in Differential Equations, 20 (2015), pp. 299360.
Abstract
In this paper we investigate linear elliptic, secondorder boundary value problems with mixed boundary conditions. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain  including a very weak compatibility condition between the Dirichlet boundary part and its complement  we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived. The essential instruments are De Giorgi and MorreyCampanato estimates. 
M. Thomas, Uniform PoincaréSobolev and relative isoperimetric inequalities for classes of domains, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 27412761.
Abstract
The aim of this paper is to prove an isoperimetric inequality relative to a ddimensional, bounded, convex domain &Omega intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(&Omega) of the center of the ball. For this, uniform Sobolev, Poincaré and PoincaréSobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d). 
N. Ahmed, G. Matthies, Higher order continuous GalerkinPetrov time stepping schemes for transient convectiondiffusionreaction equations, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), pp. 14291450.
Abstract
We present the analysis for the higher order continuous GalerkinPetrov (cGP) time discretization schemes in combination with the onelevel local projection stabilization in space applied to timedependent convectiondiffusionreaction problems. Optimal apriori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous GalerkinPetrov and discontinuous Galerkin time discretization schemes will be given. 
W. Dreyer, R. Huth, A. Mielke, J. Rehberg, M. Winkler, Global existence for a nonlocal and nonlinear FokkerPlanck equation, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 293315.
Abstract
We consider a FokkerPlanck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the timedependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state. 
M.H. Farshbaf Shaker, A relaxation approach to vectorvalued AllenCahn MPEC problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 72 (2015), pp. 325351.

M.H. Farshbaf Shaker, Ch. Heinemann, A phase field approach for optimal boundary control of damage processes in twodimensional viscoelastic media, Mathematical Models & Methods in Applied Sciences, 25 (2015), pp. 27492793.
Abstract
In this work we investigate a phase field model for damage processes in twodimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish globalintime existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility as well as boundedness of the phase field variable which results in a doubly constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model . To this end, we prove existence of minimizers and study a family of “local” approximations via adapted cost functionals. 
H. Hanke, D. Knees, Homogenization of elliptic systems with nonperiodic, state dependent coefficients, Asymptotic Analysis, 92 (2015), pp. 203234.
Abstract
In this paper, a homogenization problem for an elliptic system with nonperiodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the nonperiodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to timedependent damage models (discussed in a future paper), we provide a Γconvergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided. 
CH. Heinemann, Ch. Kraus, A degenerating CahnHilliard system coupled with complete damage processes, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 388403.
Abstract
Complete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a timedepending domain for phase separation and complete damage processes under timevarying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasistatic balance equation for the displacement field. All these equations are strongly nonlinearly coupled. Because of the doubly degenerating character and the doubly nonlinear differential inclusion we are confronted with introducing a suitable notion of weak solutions. We choose a notion of weak solutions which consists of weak formulations of the diffusion equation and the momentum balance, a onesided variational inequality for the damage function and an energy estimate. For the introduced degenerating system, we prove existence of weak solutions in an $SBV$framework. The existence result is based on an approximation system, where the elliptic degeneracy of the displacement field and the parabolic degeneracy of the concentration are eliminated. In the framework of phase separation and damage, this means that the approximation system allows only for partial damage and a nondegenerating mobility tensor. For the approximation system, existence results are established. Then, a passage to the limit shows existence of weak solutions of the degenerating system. 
CH. Heinemann, Ch. Kraus, Complete damage in linear elastic materials  Modeling, weak formulation and existence results, Calculus of Variations and Partial Differential Equations, 54 (2015), pp. 217250.
Abstract
We introduce a complete damage model with a timedepending domain for linearelastically stressed solids under timevarying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasistatic balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove localintime existence and globalintime existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a timedepending domain. In this context, two major challenges arise: Firstly, the timedependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $Gamma$convergence techniques. Secondly, the timedepending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 25652590.
Abstract
In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolicparabolic system. To this end, we first generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, timediscretization and different variational techniques. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolicparabolic phase field system with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 20442073.
Abstract
The aim of this paper is to prove existence of weak solutions of hyperbolicparabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing timediscretization, $H^2$regularization and variational techniques. 
CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 45874612.
Abstract
In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damagedependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging. 
A. Mielke, J. Naumann, Globalintime existence of weak solutions to Kolmogorov's twoequation model of turbulence, Comptes Rendus Mathematique. Academie des Sciences. Paris, 353 (2015), pp. 321326.
Abstract
We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ^{3}. This model consists in a NavierStokes type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ^{3} cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {u, ω, k} to the problem under consideration, with ω, k obeying the inequalities and . 
A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 26792700.
Abstract
We discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable onedimensional SwiftHohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the timedependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L^{2}, while for the case of a quadratic nonlinearity we need to impose weak convergence in H^{1}. However, we do not need wellpreparedness of the initial conditions. 
A. Mielke, J. Haskovec, P.A. Markowich, On uniform decay of the entropy for reactiondiffusion systems, Journal of Dynamics and Differential Equations, 27 (2015), pp. 897928.
Abstract
In this work we derive entropy decay estimates for a class of nonlinear reactiondiffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the logSobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions. 
E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 12651341.

S. Heinz, On the structure of the quasiconvex hull in planar elasticity, Calculus of Variations and Partial Differential Equations, 50 (2014), pp. 481489.
Abstract
Let K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rankone convex hull in planar elasticity. 
S. Jachalski, G. Kitavtsev, R. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014), pp. 527544.
Abstract
The existence of global nonnegative weak solutions is proved for coupled onedimen sional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navierslip or noslip conditions at both liquidliquid and liquidsolid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown. 
B. Barrios, I. Peral, F. Soria, E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Archive for Rational Mechanics and Analysis, 213 (2014), pp. 629650.

A. Cesaroni, M. Novaga, E. Valdinoci, A symmetry result for the OrnsteinUhlenbeck operator, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 24512467.

R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 144 (2014), pp. 831855.

N. Abatangelo, E. Valdinoci, A notion of nonlocal curvature, Numerical Functional Analysis and Optimization. An International Journal, 35 (2014), pp. 793815.

P. Colli, G. Gilardi, P. Krejčí, P. PodioGuidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous CahnHilliard system, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 10611087.
Abstract
In this paper we propose a time discretization of a system of two parabolic equations describing diffusiondriven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundaryvalue problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step. 
P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Mathematical Methods in the Applied Sciences, 37 (2014), pp. 13181324.
Abstract
The present note deals with a nonstandard systems of differential equations describing a twospecies phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential. 
P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014), pp. 257275.
Abstract
We are concerned with a nonstandard phase field model of CahnHilliard type. The model, which was introduced by PodioGuidugli (Ric. Mat. 2006), describes twospecies phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, PodioGuidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initialboundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution. 
P. Colli, G. Gilardi, J. Sprekels, On the CahnHilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014), pp. 972994.
Abstract
The CahnHilliard and viscous CahnHilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some wellposedness and regularity results are proved. 
M. Cozzi, A. Farina, E. Valdinoci, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Communications in Mathematical Physics, 331 (2014), pp. 189214.

M.M. Fall, E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of (Delta) su+u=up in RN when s is close to 1, Communications in Mathematical Physics, 329 (2014), pp. 383404.

A. Farina, E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calculus of Variations and Partial Differential Equations, 49 (2014), pp. 923936.

A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 94 (2014), pp. 156170.

A. Gloria, S. Neukamm, F. Otto, An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 325346.
Abstract
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author. 
R. HallerDintelmann, W. Höppner, H.Chr. Kaiser, J. Rehberg, G. Ziegler, Optimal elliptic Sobolev regularity near threedimensional, multimaterial Neumann vertices, Functional Analysis and its Applications, 48 (2014), pp. 208222.
Abstract
We investigate optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic divgrad operators in three dimensions at a multimaterial vertex on the Neumann boundary part of the polyhedral spatial domain. The gradient of a solution to the corresponding elliptic PDE (in a neighbourhood of the vertex) is integrable to an index greater than three. 
P. Hornung, S. Neukamm, I. Velcic, Derivation of a homogenized nonlinear plate theory from 3D elasticity, Calculus of Variations and Partial Differential Equations, 51 (2014), pp. 677699.

S. Melchionna, E. Rocca, On a nonlocal CahnHilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014), pp. 461497.
Abstract
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u). 
A. Miranville, E. Rocca, G. Schimperna, A. Segatti, The PenroseFife phasefield model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 42594290.

O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, Journal de Mathématiques Pures et Appliquées, 101 (2014), pp. 126.

A.F.M. TER Elst, M. Meyries, J. Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 193 (2014), pp. 12951318.
Abstract
We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable L^{p}setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems. 
D.A. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 99 (2014), pp. 4979.

A. Mielke, S. Reichelt, M. Thomas, Twoscale homogenization of nonlinear reactiondiffusion systems with slow diffusion, Networks Heterogeneous Media, 9 (2014), pp. 353382.
Abstract
We derive a twoscale homogenization limit for reactiondiffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the twoscale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit. 
A. Mielke, Ch. Ortner, Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM Journal on Mathematical Analysis, 46 (2014), pp. 13171347.
Abstract
We formulate quasistatic nonlinear finitestrain viscoelasticity of ratetype as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure timedependent frameindifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the timeincremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the onedimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the timecontinuous in some case in a specific case. 
A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the largedeviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 12931325.
Abstract
Motivated by the occurence in rate functions of timedependent largedeviation principles, we study a class of nonnegative functions ℒ that induce a flow, given by ℒ(z_{t},ż_{t})=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropyWasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure. 
A. Mielke, S. Zelik, On the vanishingviscosity limit in parabolic systems with rateindependent dissipation terms, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XIII (2014), pp. 67135.
Abstract
We consider quasilinear parabolic systems with a nonsmooth rateindependent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishingviscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishingviscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rateindependent systems. 
S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014), DOI 10.1007/s0052601407652 .
Abstract
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gammaconvergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using twoscale convergence. This is a nontrivial task, since one has to treat twoscale convergence in connection with a nonlinear differential constraint. 
P.É. Druet, Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface, Mathematica Bohemica, 138 (2013), pp. 185224.
Abstract
We investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of squareintegrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated. 
CH. Heinemann, Ch. Kraus, Existence results for diffuse interface models describing phase separation and damage, European Journal of Applied Mathematics, 24 (2013), pp. 179211.
Abstract
In this paper we analytically investigate CahnHilliard and AllenCahn systems which are coupled with elasticity and unidirectional damage processes. We are interested in the case where the free energy contains logarithmic terms of the chemical concentration variable and quadratic terms of the gradient of the damage variable. For elastic CahnHilliard and AllenCahn systems coupled with unidirectional damage processes, an appropriate notion of weak solutions is presented as well as an existence result based on certain regularization methods and an higher integrability result for the strain. 
K. Götze, Strong solutions for the interaction of a rigid body and a viscoelastic fluid, Journal of Mathematical Fluid Mechanics, 15 (2013), pp. 663688.
Abstract
We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique localintime strong solution exists. 
M. Liero, U. Stefanelli, A new minimum principle for Lagrangian mechanics, Journal of Nonlinear Science, 23 (2013), pp. 179204.
Abstract
We present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameterdependent globalintime minimization problems is presented and the convergence of the respective minimizers to the solution of the system of Lagrange's equations is ascertained. Moreover, we extend this perspective to mixed dissipative/nondissipative situations, present a finite timehorizon version of this approach, and provide related Gammaconvergence results. Finally, some discussion on corresponding timediscrete versions of the principle is presented. 
M. Liero, U. Stefanelli, Weighted InertiaDissipationEnergy functionals for semilinear equations, Bollettino della Unione Matematica Italiana. Serie 9, VI (2013), pp. 127.

M. Liero, A. Mielke, Gradient structures and geodesic convexity for reactiondiffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120346/120120346/28.
Abstract
We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. 
M. Liero, Passing from bulk to bulk/surface evolution in the AllenCahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013), pp. 919942.
Abstract
In this paper we formulate a boundary layer approximation for an AllenCahntype equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2gradient flow with the corresponding AllenCahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions. 
S. Neukamm, I. Velcic, Derivation of a homogenized vonKármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 27012748.
Abstract
We rigorously derive a homogenized vonKármán plate theory as a Gammalimit from nonlinear threedimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, threedimensional plate with spatially periodic material properties. The functional features two small length scales: the period $epsilon$ of the elastic composite material, and the thickness h of the slender plate. We study the behavior as $epsilon$ and h simultaneously converge to zero in the vonKármán scaling regime. The obtained limit is a homogenized vonKármán plate model. Its effective material properties are determined by a relaxation formula that exposes a nontrivial coupling of the behavior of the outofplane displacement with the oscillatory behavior in the inplane directions. In particular, the homogenized coefficients depend on the relative scaling between h and $epsilon$, and different values arise for h<<$epsilon$, $epsilon$ h and $epsilon$ << h. 
A. Fiaschi, D. Knees, S. Reichelt, Global higher integrability of minimizers of variational problems with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 401 (2013), pp. 269288.
Abstract
We consider integral functionals with densities of pgrowth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of pgrowth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, An asymptotic analysis for a nonstandard CahnHilliard system with viscosity, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 353368.
Abstract
This paper is concerned with a diffusion model of phasefield type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term  respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$  with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is wellposed and investigated the longtime behavior of its $(varepsilon,delta)$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)$solutions to the corresponding solutions for the case $eps$ =0, whose longtime behavior we characterize; in the proofs, we employ compactness and monotonicity arguments. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate CahnHilliard system with viscosity, Journal of Differential Equations, 254 (2013), pp. 42174244.
Abstract
Existence and uniqueness are investigated for a nonlinear diffusion problem of phasefield type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model twospecies phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phasefield equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a globalintime solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables. 
M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of manyparticle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 11661188.

M. Geissert, K. Götze, M. Hieber, $L^p$theory for strong solutions to fluidrigid body interaction in Newtonian and generalized Newtonian fluids, Transactions of the American Mathematical Society, 365 (2013), pp. 13931439.
Abstract
Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shearthinning or shearthickening fluids of powerlaw type of exponent $ dgeq 1$. We develop a method to prove that this system admits a unique, local, strong solution in the $ L^p$setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasilinear evolution equations and requires that the exponent $ p$ satisfies the condition $ p>5$. 
M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the FokkerPlanck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/11350017/43.

A. Glitzky, A. Mielke, A gradient structure for systems coupling reactiondiffusion effects in bulk and interfaces, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 64 (2013), pp. 2952.
Abstract
We derive gradientflow formulations for systems describing driftdiffusion processes of a finite number of species which undergo massaction type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electroreactiondiffusion systems with active interfaces permitting driftdiffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the selfconsistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models. 
D. Knees, R. Rossi, Ch. Zanini, A vanishing viscosity approach to a rateindependent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 565616.
Abstract
We analyze a rateindependent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the bynow classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rateindependent damage models as limits of systems driven by viscous, ratedependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparameterization. In this way, in the limit we obtain a novel formulation for the rateindependent damage model, which highlights the interplay of viscous and rateindependent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps. 
D. Knees, A. Schröder, Computational aspects of quasistatic crack propagation, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 6399.
Abstract
The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rateindependent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with selfcontact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations. 
M. Thomas, Quasistatic damage evolution with spatial BVregularization, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 235255.
Abstract
An existence result for energetic solutions of rateindependent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W^{1,r}(Ω) with r∈(1,∞) for Ω⊂R^{d}. We now cover the case r=1. The lack of compactness in W^{1,1}(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γlimit of functionals of ModicaMortola type. 
D. Hömberg, K. Krumbiegel, J. Rehberg, Optimal control of a parabolic equation with dynamic boundary condition, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013), pp. 331.
Abstract
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions. 
A. Mielke, R. Rossi, G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calculus of Variations and Partial Differential Equations, 46 (2013), pp. 253310.
Abstract
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a timediscretization scheme with variational techniques. Finally, we discuss an application to a material model in finitestrain elasticity. 
A. Mielke, E. Rohan, Homogenization of elastic waves in fluidsaturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 873916.
Abstract
We consider periodically heterogeneous fluidsaturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the twoscale homogenization method we obtain the limit twoscale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena. 
A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), pp. 131.
Abstract
We consider finitedimensional, timecontinuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λconvexity of the entropy and treat several examples including some more general nonlinear reaction systems 
A. Mielke, Thermomechanical modeling of energyreactiondiffusion systems, including bulkinterface interactions, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 479499.
Abstract
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes AllenCahn, CahnHilliard, and reactiondiffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,Φ,K) defining the evolution $dot U$=  K(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K^{1} exists, the triple (X,Φ,G) defines a gradient system. Onsager systems are well suited to model bulkinterface interactions by using the dual dissipation potential Ψ^{*}(U, Ξ)= ½ ⟨Ξ K(U) Ξ⟩. Then, the two functionals Φ and Ψ^{*} can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts. 
P.E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 69 (2012), pp. 233258.
Abstract
In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution. Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higherorder estimates, and for the existence result. 
M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012), pp. 437457.
Abstract
This paper deals with dimension reduction in linearized elastoplasticity in the rateindependent case. The reference configuration of the elastoplastic body is given by a twodimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rateindependent evolution formulated in the framework of energetic solutions. This concept is based on an energystorage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance. 
A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasistatic damage evolution, Archive for Rational Mechanics and Analysis, 203 (2012), pp. 415453.

S. Arnrich, A. Mielke, M.A. Peletier, G. Savaré, M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calculus of Variations and Partial Differential Equations, 44 (2012), pp. 419454.
Abstract
We study a singularlimit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):18051825, 2010, using the linear structure of the equation. In this paper we reprove the result by using solely the Wasserstein gradientflow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a secondorder system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradientflow structure, we prove that the sequence of rescaled solutions is precompact in an appropriate topology. We then prove a Gammaconvergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $e$problem converge to a solution of the limiting problem. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Continuous dependence for a nonstandard CahnHilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012), pp. 2752.
Abstract
This note is concerned with a nonlinear diffusion problem of phasefield type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of twospecies phase segregation on an atomic lattice [PodioGuidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/PodioGuidugli/Sprekels 2012. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 437459.
Abstract
We investigate a distributed optimal control problem for a phase field model of CahnHilliard type. The model describes twospecies phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the firstorder necessary conditions of optimality. 
P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Global existence for a strongly coupled CahnHilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012), pp. 495513.
Abstract
An existence result is proved for a nonlinear diffusion problem of phasefield type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model twospecies phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phasefield equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables. 
P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan Journal of Mathematics, 80 (2012), pp. 119149.
Abstract
We investigate a nonstandard phase field model of CahnHilliard type. The model, which was introduced in PodioGuidugli (2006), describes twospecies phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, PodioGuidugli, and Sprekels (2011a and b) for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show wellposedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the firstorder necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type. 
K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finitestrain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012), pp. 888909.
Abstract
We study the time evolution in elastoplasticity within the rateindependent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of timeincremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented. 
A.F.M. TER Elst, J. Rehberg, $L^infty$estimates for divergence operators on bad domains, Analysis and Applications, 10 (2012), pp. 207214.
Abstract
In this paper, we prove $L^infty$estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it. 
G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasiincompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012), pp. 5477.
Abstract
In this contribution, we investigate a diffuse interface model for quasiincompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective CahnHilliard equation numerically by a Local Discontinuous Galerkin scheme. 
A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012), pp. 38743900.
Abstract
We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generationrecombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level. 
D. Knees, A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 18591884.
Abstract
A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B^{ 3/2 }_{ 2,∞} regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FEdiscretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results. 
W. Dreyer, J. Giesselmann, Ch. Kraus, Ch. Rohde, Asymptotic analysis for Korteweg models, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 14 (2012), pp. 105143.
Abstract
This paper deals with a sharp interface limit of the isothermal NavierStokesKorteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models. 
A. Mielke, R. Rossi, G. Savaré, Variational convergence of gradient flows and rateindependent evolutions in metric spaces, Milan Journal of Mathematics, 80 (2012), pp. 381410.
Abstract
We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metricdissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blowup by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rateindependent problems arise naturally as a limit of pgradient flows, p>1, when the exponents p converge to 1. 
A. Mielke, Emergence of rateindependent dissipation from viscous systems with wiggly energies, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 591606.
Abstract
We consider the passage from viscous system to rateindependent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R^{*}) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for CahnHilliard systems coupled with elasticity and damage, Advances in Mathematical Sciences and Applications, 21 (2011), pp. 321359.
Abstract
The CahnHilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the socalled CahnLarché system by combining it with elasticity to capture nonneglecting deformation phenomena, which occurs during phase separation in the material. Evolving microstructures such as phase separation and coarsening processes have a strong influence on damage initiation and propagation in alloys. We develop the existing framework of CahnHilliard and CahnLarché systems by coupling these systems with a unidirectional evolution inclusion for an internal variable, describing damage processes. After establishing a weak notion of the corresponding evolutionary system, we prove existence of weak solutions for ratedependent damage processes under certain growth conditions of the energy functional. 
P. Colli, P. Krejčí, E. Rocca, J. Sprekels, A nonlocal quasilinear multiphase system with nonconstant specific heat and heat conductivity, Journal of Differential Equations, 251 (2011), pp. 13541387.

P. Colli, G. Gilardi, P. PodioGuidugli, J. Sprekels, Wellposedness and longtime behavior for a nonstandard viscous CahnHilliard system, SIAM Journal on Applied Mathematics, 71 (2011), pp. 18491870.
Abstract
We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a globalintime smooth solution to the associated initial/boundaryvalue problem; moreover, we give a description of the relative $omega$limit set. 
R. HallerDintelmann, H.Chr. Kaiser, J. Rehberg, Direct computation of elliptic singularities across anisotropic, multimaterial edges, Journal of Mathematical Sciences (New York), 172 (2011), pp. 589622.
Abstract
We characterise the singularities of elliptic divgrad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two or threedimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three and fourmaterial edges. These bounds can be used to prove optimal regularity results for elliptic divgrad operators on threedimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark Lshape problem. 
K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the NavierStokesKorteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239254.
Abstract
In this contribution we will study the behaviour of the pressure across phase boundaries in liquidvapour flows. As mathematical model we will consider the static version of the NavierStokesKorteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations. 
A. Glitzky, Analysis of electronic models for solar cells including energy resolved defect densities, Mathematical Methods in the Applied Sciences, 34 (2011), pp. 19801998.
Abstract
We introduce an electronic model for solar cells including energy resolved defect densities. The resulting driftdiffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ODEs containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium the free energy along solutions decays monotonously. In other cases it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is done by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level. 
CH. Kraus, The degenerate and nondegenerate Stefan problem with inhomogeneous and anisotropic GibbsThomson law, European Journal of Applied Mathematics, 22 (2011), pp. 393422.
Abstract
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic GibbsThomson condition at the phase boundary. We show the longtime existence of weak solutions for the nondegenerate Stefan problem with a spatially inhomogeneous and anisotropic GibbsThomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic GibbsThomson law in a weak generalized BVformulation. 
A. Mielke, U. Stefanelli, Weighted energydissipation functionals for gradient flows, ESAIM. Control, Optimisation and Calculus of Variations, 17 (2011), pp. 5285.
Abstract
We investigate a globalintime variational approach to abstract evolution by means of the weighted energydissipation functionals proposed by Mielke & Ortiz in “A class of minimum principles for characterizing the trajectories of dissipative systems”. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with timediscretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from S. Conti and M. Ortiz “Minimum principles for the trajectories of systems governed by rate problems”. 
A. Mielke, A gradient structure for reactiondiffusion systems and for energydriftdiffusion systems, Nonlinearity, 24 (2011), pp. 13291346.
Abstract
In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reactiondiffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. 
A. Mielke, Completedamage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems  Series S, 4 (2011), pp. 423439.
Abstract
The rateindependent damage model recently developed in Bouchitté, Mielke, Roubíček “A completedamage problem at small strains" allows for complete damage, such that the deformation is no longer welldefined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but nonquadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions. 
A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems, Mathematische Nachrichten, 284 (2011), pp. 21592174.
Abstract
Our focus are energy estimates for discretized reactiondiffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete SobolevPoincaré inequality. 
J.A. Griepentrog, L. Recke, Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems, Journal of Evolution Equations, 10 (2010), pp. 341375.
Abstract
A general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in SobolevMorrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data. 
M. Thomas, A. Mielke, Damage of nonlinearly elastic materials at small strain  Existence and regularity results, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 88112.
Abstract
In this paper an existence result for energetic solutions of rateindependent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [MielkeRoubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W^1,r (Omega) with r>d for Omega ⊂ R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz and Höldercontinuity of solutions with respect to time. 
H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified GibbsThomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511545.
Abstract
We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified GibbsThomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the EulerLagrange equation of the diffuse phase field energy. 
J. Giannoulis, A. Mielke, Ch. Sparber, Highfrequency averaging in semiclassical Hartreetype equations, Asymptotic Analysis, 70 (2010), pp. 87100.
Abstract
We investigate the asymptotic behavior of solutions to semiclassical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a highfrequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras. 
P. Gruber, D. Knees, S. Nesenenko, M. Thomas, Analytical and numerical aspects of timedependent models with internal variables, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), pp. 861902.
Abstract
In this paper some analytical and numerical aspects of timedependent models with internal variables are discussed. The focus lies on elasto/viscoplastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elastoplasticity with hardening and viscous models of the NortonHoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rateindependent processes is explained and temporal regularity results based on different convexity assumptions are presented. 
R. HallerDintelmann, J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains, Archiv der Mathematik, 95 (2010), pp. 457468.
Abstract
We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of nonzero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^1,2$ are identified as positive measures. 
W. Dreyer, Ch. Kraus, On the van der WaalsCahnHilliard phasefield model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 11611186.
Abstract
We study the equilibria of liquidvapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der WaalsCahnHilliard theory.
For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the twophase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density. 
A. Glitzky, K. Gärtner, Existence of bounded steady state solutions to spinpolarized driftdiffusion systems, SIAM Journal on Mathematical Analysis, 41 (2010), pp. 24892513.
Abstract
We study a stationary spinpolarized driftdiffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a ScharfetterGummel scheme on 3D boundary conforming Delaunay grids we show existence, boundedness and, for small applied voltages, the uniqueness of the discrete solution. 
D. Hömberg, Ch. Meyer, J. Rehberg, W. Ring, Optimal control for the thermistor problem, SIAM Journal on Control and Optimization, 48 (2010), pp. 34493481.
Abstract
This paper is concerned with the stateconstrained optimal control of the twodimensional thermistor problem, a quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive firstorder necessary conditions for the optimal control problem. 
H.Chr. Kaiser, J. Rehberg, Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge, Journal of Mathematical Sciences (New York), 169 (2010), pp. 145166.
Abstract
We investigate optimal elliptic regularity of anisotropic divgrad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces. 
D. Knees, On global spatial regularity and convergence rates for time dependent elastoplasticity, Mathematical Models & Methods in Applied Sciences, 20 (2010), pp. 18231858.

A. Mielke, T. Roubíček, J. Zeman, Complete damage in elastic and viscoelastic media and its energetics, Computer Methods in Applied Mechanics and Engineering, 199 (2010), pp. 12421253.
Abstract
A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The “flow rule” for the damage parameter is rateindependent. The stored energy involves the gradient of the damage variable, which determines an internal lengthscale. Quasistatic fully rateindependent evolution is considered as well as ratedependent evolution including viscous/inertial effects. Illustrative 2dimensional computer simulations are presented, too. 
J. Sprekels, H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 72 (2010), pp. 30283048.

K. Hoke, H.Chr. Kaiser, J. Rehberg, Analyticity for some operator functions from statistical quantum mechanics, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 10 (2009), pp. 749771.
Abstract
For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with nonsmooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to threedimensional Lipschitz domain factorizes over the space of essentially bounded functions. 
A. Petrov, M. Schatzman, Mathematical results on existence for viscoelastodynamic problems with unilateral constraints, SIAM Journal on Mathematical Analysis, 40 (2009), pp. 18821904.
Abstract
We study a damped wave equation and the evolution of a KelvinVoigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution. 
G. Bouchitté, A. Mielke, T. Roubíček, A completedamage problem at small strains, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 205236.
Abstract
The complete damage of a linearlyresponding material that can thus completely disintegrate is addressed at small strains under timevarying Dirichlet boundary conditions as a rateindependent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a onedimensional example. 
R. HallerDintelmann, Ch. Meyer, J. Rehberg, A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 60 (2009), pp. 397428.
Abstract
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is reestablished for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented. 
R. HallerDintelmann, J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, Journal of Differential Equations, 247 (2009), pp. 13541396.
Abstract
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 
A. Mainik, A. Mielke, Global existence for rateindependent gradient plasticity at finite strain, Journal of Nonlinear Science, 19 (2009), pp. 221248.
Abstract
We provide a global existence result for the timecontinuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables. 
S. Zelik, A. Mielke, Multipulse evolution and spacetime chaos in dissipative systems, Memoirs of the American Mathematical Society, 198 (2009), pp. 197.

A. Glitzky, Energy estimates for electroreactiondiffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 159174.
Abstract
We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electroreactiondiffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discretetime version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model. 
A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electroreactiondiffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), pp. 788805.
Abstract
We consider electroreactiondiffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations.
We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electroreactiondiffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. 
H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of driftdiffusion equations for semiconductor devices: The 2D case, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009), pp. 15841605.
Abstract
We regard driftdiffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasilinear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. This investigation puts special emphasis on nonsmooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation. 
H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Monotonicity properties of the quantum mechanical particle density: An elementary proof, Monatshefte fur Mathematik, 158 (2009), pp. 179185.
Abstract
An elementary proof of the antimonotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included. 
D. Knees, On global spatial regularity in elastoplasticity with linear hardening, Calculus of Variations and Partial Differential Equations, 36 (2009), pp. 611625.
Abstract
We study the global spatial regularity of solutions of elastoplastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L^{∞}((0,T);H^{3/2δ}(Ω)) for the displacements and z in L^{∞}((0,T);H^{1/2δ}(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument. 
P. Krejčí, M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009), pp. 117145.
Abstract
The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one. 
A. Mielke, F. Rindler, Reverse approximation of energetic solutions to rateindependent processes, NoDEA. Nonlinear Differential Equations and Applications, 16 (2009), pp. 1740.
Abstract
Energetic solutions to rateindependent processes are usually constructed via timeincremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit. 
A. Mielke, R. Rossi, G. Savaré, Modeling solutions with jumps for rateindependent systems on metric spaces, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 585615.
Abstract
Rateindependent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rateindependent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given. 
A. Mielke, T. Roubíček, Numerical approaches to rateindependent processes and applications in inelasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), pp. 399429.
Abstract
A general abstract approximation scheme for rateindependent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shapememory alloys. 
A. Mielke, L. Paoli, A. Petrov, On existence and approximation for a 3D model of thermallyinduced phase transformations in shapememory alloys, SIAM Journal on Mathematical Analysis, 41 (2009), pp. 13881414.
Abstract
This paper deals with a threedimensional model for thermal stressinduced transformations in shapememory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rateindependent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also spacetime discretizations and establish convergence of these approximations. 
H. Neidhardt, V.A. Zagrebnov, Linear nonautonomous Cauchy problems and evolution semigroups, Advances in Differential Equations, 14 (2009), pp. 289340.
Abstract
The paper is devoted to the problem of existence of propagators for an abstract linear nonautonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the socalled evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear nonautonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to timedependent Schrödinger operators with moving point interactions in 1D. 
H. Stephan, Modeling of driftdiffusion systems, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 3353.
Abstract
We derive driftdiffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, nondegenerate diffusivity matrices turn out to be diagonal, or  in the case of cross diffusion  even constant. 
P.É. Druet, Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and righthand side in $L^p$ with $pge 1$, Mathematical Methods in the Applied Sciences, 32 (2008), pp. 135166.
Abstract
Accurate modeling of heat transfer in hightemperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a righthand side in Lp with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and righthand side in Lp with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the Lpnorm of the heatsources, and to pass to the limit. 
J.A. Griepentrog, W. Höppner, H.Chr. Kaiser, J. Rehberg, A biLipschitz continuous, volume preserving map from the unit ball onto a cube, Note di Matematica, 28 (2008), pp. 185201.
Abstract
We construct two biLipschitz, volume preserving maps from Euclidean space onto itself which map the unit ball onto a cylinder and onto a cube, respectively. Moreover, we characterize invariant sets of these mappings. 
S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), pp. 795801.

F. Auricchio, A. Mielke, U. Stefanelli, A rateindependent model for the isothermal quasistatic evolution of shapememory materials, Mathematical Methods in the Applied Sciences, 18 (2008), pp. 125164.
Abstract
This note addresses a threedimensional model for isothermal stressinduced transformation in shapememory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rateindependent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasistatic evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics. 
H. Cornean, K. Hoke, H. Neidhardt, P.N. Racec, J. Rehberg, A KohnSham system at zero temperature, Journal of Physics. A. Mathematical and General, 41 (2008), pp. 385304/1385304/21.
Abstract
An onedimensional KohnSham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective KohnSham potential and obtain $W^1,2$bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchangecorrelation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero. 
R. HallerDintelmann, H.Chr. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées, 89 (2008), pp. 2548.

M. Hieber, J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 40 (2008), pp. 292305.
Abstract
In this paper we investigate quasilinear systems of reactiondiffusion equations with mixed DirichletNeumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heatkernel estimates we prove existence of a unique solution to systems of this type. 
R. Rossi, A. Mielke, G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, VII (2008), pp. 97169.
Abstract
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces. 
J.C. DE Los Reyes, P. Merino, J. Rehberg, F. Tröltzsch, Optimality conditions for stateconstrained PDE control problems with finitedimensional control space, Control and Cybernetics, 37 (2008), pp. 738.

A. Glitzky, R. Hünlich, Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains, Mathematische Nachrichten, 281 (2008), pp. 16761693.
Abstract
We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the twodimensional stationary energy model. For this purpose we derive a $W^1,p$regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. 
A. Glitzky, Analysis of a spinpolarized driftdiffusion model, Advances in Mathematical Sciences and Applications, 18 (2008), pp. 401427.
Abstract
We introduce a spinpolarized driftdiffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. We give a weak formulation of this problem and prove an existence and uniqueness result for the instationary problem. If the boundary data is compatible with thermodynamic equilibrium the free energy along the solution decays monotonously and exponentially to its equilibrium value. In other cases it may be increasing but we estimate its growth. Moreover we give upper and lower estimates for the solution. 
A. Glitzky, Exponential decay of the free energy for discretized electroreactiondiffusion systems, Nonlinearity, 21 (2008), pp. 19892009.
Abstract
Our focus are electroreactiondiffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electroreactiondiffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. 
D. Knees, P. Neff, Regularity up to the boundary for nonlinear elliptic systems arising in timeincremental infinitesimal elastoplasticity, SIAM Journal on Mathematical Analysis, 40 (2008), pp. 2143.
Abstract
In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the timediscretization of models from infinitesimal elastoplasticity. Our main focus lies on an elastoplastic Cosserat model. More specifically we show that the time discretization renders $H^2$regularity of the displacement and $H^1$regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and PrandtlReuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elastoplasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (LegendreHadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the meshwidth $h$ in error estimates. 
D. Knees, A. Mielke, Energy release rate for cracks in finitestrain elasticity, Mathematical Methods in the Applied Sciences, 31 (2008), pp. 501528.
Abstract
Griffith's fracture criterion describes in a quasistatic setting whether or not a preexisting crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the wellknown Griffith formula and the $J$integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. 
D. Knees, A. Mielke, Ch. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models & Methods in Applied Sciences, 18 (2008), pp. 15291569.
Abstract
We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions. 
D. Knees, A. Mielke, On the energy release rate in finitestrain elasticity, Mechanics of Advanced Materials and Structures, 15 (2008), pp. 421427.

D. Knees, Global stress regularity of convex and some nonconvex variational problems, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 187 (2008), pp. 157184.

A. Mielke, Weakconvergence methods for Hamiltonian multiscale problems, Discrete and Continuous Dynamical Systems, 20 (2008), pp. 5379.
Abstract
We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the welldeveloped theory of $Gamma$convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients 
A. Mielke, A. Petrov, J.A.C. Martins, Convergence of solutions of kinetic variational inequalities in the rateindependent quasistatic limit, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 10121020.
Abstract
This paper discusses the convergence of kinetic variational inequalities to rateindependent quasistatic variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rateindependent quasistatic problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rateindependent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to threedimensional elasticplastic systems with hardening is given. 
J.A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in SobolevMorrey spaces, Advances in Differential Equations, 12 (2007), pp. 10311078.
Abstract
This text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEVMORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER continuous in time and space up to the boundary for a certain range of MORREY exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients. 
J.A. Griepentrog, SobolevMorrey spaces associated with evolution equations, Advances in Differential Equations, 12 (2007), pp. 781840.
Abstract
In this text we introduce new classes of SOBOLEVMORREY spaces being adequate for the regularity theory of second order parabolic boundary value problems on LIPSCHITZ domains of space dimension greater or equal than three with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, LIPSCHITZ transformation, and reflection. In the second part of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned SOBOLEVMORREY spaces of solutions and right hand sides. 
J. Elschner, H.Chr. Kaiser, J. Rehberg, G. Schmidt, $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models & Methods in Applied Sciences, 17 (2007), pp. 593615.

J. Elschner, J. Rehberg, G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 9 (2007), pp. 233252.
Abstract
We prove an optimal regularity result for elliptic operators $nabla cdot mu nabla:W^1,q_0 rightarrow W^1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators. 
A. Glitzky, R. Hünlich, Resolvent estimates in $W^1,p$ related to strongly coupled linear parabolic systems with coupled nonsmooth capacities, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 22152232.
Abstract
We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in $W^1,p$ spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain $L^p$ estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. 
A. Mielke, A. Petrov, Thermally driven phase transformation in shapememory alloys, Advances in Mathematical Sciences and Applications, 17 (2007), pp. 667685.
Abstract
This paper analyzes a model for phase transformation in shapememory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rateindependent processes. Existence and uniqueness results are proved. 
A. Mielke, R. Rossi, Existence and uniqueness results for a class of rateindependent hysteresis problems, Mathematical Models & Methods in Applied Sciences, 17 (2007), pp. 81123.

A. Mielke, A. Timofte, Twoscale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007), pp. 642668.
Abstract
This paper is devoted to the twoscale homogenization for a class of rateindependent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energystorage functional and a dissipation functional. Using the recently developed method of weak and strong twoscale convergence via periodic unfolding, we show that these two functionals have a suitable twoscale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using socalled joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals. 
A. Mielke, S. Zelik, Infinitedimensional hyperbolic sets and spatiotemporal chaos in reactiondiffusion systems in $R^n$, Journal of Dynamics and Differential Equations, 19 (2007), pp. 333389.

A. Mielke, A model for temperatureinduced phase transformations in finitestrain elasticity, IMA Journal of Applied Mathematics, 72 (2007), pp. 644658.

F. Schmid, A. Mielke, Existence results for a contact problem with varying friction coefficient and nonlinear forces, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 616631.
Abstract
We consider the rateindependent problem of a particle moving in a three  dimensional half space subject to a timedependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rateindependent systems as developed by Mielke and coworkers. We generalize the timeincremental minimization procedure of Mielke and Rossi for the present situation of a nonassociative flow rule. 
J. Giannoulis, A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), pp. 493523.

M. Kočvara, A. Mielke, T. Roubíček, A rateindependent approach to the delamination problem, Mathematics and Mechanics of Solids, 11 (2006), pp. 423447.

V. Pata, S. Zelik, A remark on the weakly damped wave equation, Communications on Pure and Applied Analysis, 5 (2006), pp. 609614.

V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), pp. 14951506.

H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA. Nonlinear Differential Equations and Applications, 13 (2006), pp. 287310.

D. Knees, Global regularity of the elastic fields of a powerlow model on Lipschitz domains, Mathematical Methods in the Applied Sciences, 29 (2006), pp. 13631391.

A. Mielke, G. Francfort, Existence results for a class of rateindependent material models with nonconvex elastic energies, Journal fur die Reine und Angewandte Mathematik, 595 (2006), pp. 5591.

A. Mielke, T. Roubíček, Rateindependent damage processes in nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 16 (2006), pp. 177209.

A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via WignerHusimi transforms, Archive for Rational Mechanics and Analysis, 181 (2006), pp. 401448.

M. Baro, H. Neidhardt, J. Rehberg, Current coupling of driftdiffusion models and dissipative SchrödingerPoisson systems: Dissipative hybrid models, SIAM Journal on Mathematical Analysis, 37 (2005), pp. 941981.

H. Neidhardt, J. Rehberg, Uniqueness for dissipative SchrödingerPoisson systems, Journal of Mathematical Physics, 46 (2005), pp. 113513/1113513/28.

M. Kružík, A. Mielke, T. Roubíček, Modelling of microstructure and its evolution in shapememoryalloy singlecrystals, in particular in CuAINi, Meccanica. International Journal of the Italian Association of Theoretical and Applied Mechanics, 40 (2005), pp. 389418.

N. Nefedov, M. Radziunas, K.R. Schneider, A. Vasil'eva, Change of the type of contrast structures in parabolic Neumann problems, Computational Mathematics and Mathematical Physics, 45 (2005), pp. 3751.

W. Dreyer, B. Wagner, Sharpinterface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005), pp. 199227.

A. Glitzky, R. Hünlich, Global existence result for pair diffusion models, SIAM Journal on Mathematical Analysis, 36 (2005), pp. 12001225.

A. Glitzky, R. Hünlich, Stationary energy models for semiconductor devices with incompletely ionized impurities, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 85 (2005), pp. 778792.

A. Mielke, A.L. Afendikov, Dynamical properties of spatially nondecaying 2D NavierStokes flows with Kolmogorov forcing in an infinite strip, Journal of Mathematical Fluid Mechanics, 7 (2005), pp. 5167.

J. Rehberg, Quasilinear parabolic equations in $L^p$, Progress in Nonlinear Differential Equations and their Applications, 64 (2005), pp. 413419.

B. Wagner, An asymptotic approach to secondkind similarity solutions of the modified porous medium equation, Journal of Engineering Mathematics, 53 (2005), pp. 201220.

M. Baro, H.Chr. Kaiser, H. Neidhardt, J. Rehberg, A quantum transmitting SchrödingerPoisson system, Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics, 16 (2004), pp. 281330.

M. Baro, H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Dissipative SchrödingerPoisson systems, Journal of Mathematical Physics, 45 (2004), pp. 2143.

J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004), pp. 153164.

V. Maz'ya, J. Elschner, J. Rehberg, G. Schmidt, Solutions for quasilinear nonsmooth evolution systems in $L^p$, Archive for Rational Mechanics and Analysis, 171 (2004), pp. 219262.

H. Gajewski, I.V. Skrypnik, On unique solvability of nonlocal driftdiffusiontype problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 56 (2004), pp. 803830.

H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 10 (2004), pp. 315336.

A. Glitzky, W. Merz, Single dopant diffusion in semiconductor technology, Mathematical Methods in the Applied Sciences, 27 (2004), pp. 133154.

A. Glitzky, R. Hünlich, Stationary solutions of twodimensional heterogeneous energy models with multiple species, Banach Center Publications, 66 (2004), pp. 135151.

A. Glitzky, Electroreactiondiffusion systems with nonlocal constraints, Mathematische Nachrichten, 277 (2004), pp. 1446.

H. Gajewski, K. Zacharias, On a nonlocal phase separation model, Journal of Mathematical Analysis and Applications, 286 (2003), pp. 1131.

H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear ellipticparabolic equations, Journal of Evolution Equations, 3 (2003), pp. 247281.

H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 52 (2003), pp. 291304.

M.A. Efendiev, H. Gajewski, S. Zelik, The finite dimensional attractor for a 4th order system of CahnHilliard type with a supercritical nonlinearity, Advances in Differential Equations, 7 (2002), pp. 10731100.

G. Albinus, H. Gajewski, R. Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), pp. 367383.

J.A. Griepentrog, K. Gröger, H.Chr. Kaiser, J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Mathematische Nachrichten, 241 (2002), pp. 110120.

J.A. Griepentrog, Linear elliptic boundary value problems with nonsmooth data: Campanato spaces of functionals, Mathematische Nachrichten, 243 (2002), pp. 1942.

A. Glitzky, R. Hünlich, Global properties of pair diffusion models, Advances in Mathematical Sciences and Applications, 11 (2001), pp. 293321.

J.A. Griepentrog, H.Chr. Kaiser, J. Rehberg, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on $Lsp p$, Advances in Mathematical Sciences and Applications, 11 (2001), pp. 87112.

W. Merz, A. Glitzky, R. Hünlich, K. Pulverer, Strong solutions for pair diffusion models in homogeneous semiconductors, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 2 (2001), pp. 541567.

I.V. Skrypnik, H. Gajewski, On the uniqueness of solution to nonlinear elliptic problem (in Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukraini. Matematika. Prirodoznavstvo. Tekhnichni Nauki, (2001), pp. 2832.

J.A. Griepentrog, L. Recke, Linear elliptic boundary value problems with nonsmooth data: Normal solvability on SobolevCampanato spaces, Mathematische Nachrichten, 225 (2001), pp. 3974.

A. Glitzky, R. Hünlich, Electroreactiondiffusion systems including cluster reactions of higher order, Mathematische Nachrichten, 216 (2000), pp. 95118.

W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000), pp. 38413871.

H.Chr. Kaiser, J. Rehberg, About a stationary SchrödingerPoisson system with KohnSham potential in a bounded two or threedimensional domain, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 41 (2000), pp. 3372.

J.A. Griepentrog, An application of the Implicit Function Theorem to an energy model of the semiconductor theory, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 79 (1999), pp. 4351.
Abstract
In this paper we deal with a mathematical model for the description of heat conduction and carrier transport in semiconductor heterostructures. We solve a coupled system of nonlinear elliptic differential equations consisting of the heat equation with Joule heating as a source, the Poisson equation for the electric field an driftdiffusion equations with temperature dependent coefficients describing the charge and current conservation, subject to general thermal and electrical boundary conditions. We prove the existence and uniqueness of Holder continuous weak solutions near thermodynamic equilibria points using the Implicit Function Theorem. To show the differentiability of maps corresponding to the weak formulation of the problem we use regularity results from the theory of nonsmooth linear elliptic boundary value problems in SobolevCampanato spaces.
Contributions to Collected Editions

A. Stephan, EDP convergence for nonlinear fastslow reaction systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 14561459, DOI 10.4171/OWR/2020/29 .

R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 89119, DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general CahnHilliard systems with fractional operators and double obstacle potentials, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 243271, DOI 10.3934/dcdss.2020213 .
Abstract
Recently, the authors derived wellposedness and regularity results for general evolutionary operator equations having the structure of a CahnHilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible doublewell potentials driving the phase separation process modeled by the CahnHilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and firstorder optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the socalled “deep quench” method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful firstorder necessary optimality conditions. 
TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 395425, DOI 10.3934/dcdss.2020345 .
Abstract
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelinde Donder kinetics. 
K. Hopf, Global existence analysis of energyreactiondiffusion systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 14181421, DOI 10.4171/OWR/2020/29 .

P. Colli, G. Gilardi, J. Sprekels, Nonlocal phase field models of viscous CahnHilliard type, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 71100, DOI 10.1007/9783030331160 .
Abstract
A nonlocal phase field model of viscous CahnHilliard type is considered. This model constitutes a nonlocal version of a model for twospecies phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. PodioGuidugli and the present authors. The resulting system of differential equations consists of a highly nonlinear parabolic equation coupled to a nonlocal ordinary differential equation, which has singular terms that render the analysis difficult. Some results are presented on the wellposedness and stability of the system as well as on the distributed optimal control problem. 
S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$regularization, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179203, DOI 10.1007/9783319759401_9 .
Abstract
We address a model for rateindependent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BVregularization. Discrete solutions are obtained using an alternate timediscrete scheme and the VariableADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rateindependent system. Moreover, we present our numerical results for two benchmark problems. 
P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous CahnHilliard system with dynamic boundary conditions, in: Proceedings of the INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 217242, DOI 10.1007/9783319759401_11 .
Abstract
This note is concerned with a nonlinear diffusion problem of phasefield type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of twospecies phase segregation on an atomic lattice and was introduced by PodioGuidugli in Ric. Mat. 55 (2006), pp.105118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the LaplaceBeltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the longtime behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omegalimit set in both cases. 
M. Thomas, A comparison of delamination models: Modeling, properties, and applications, in: Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, P. VAN Meurs, M. Kimura, H. Notsu, eds., 30 of Mathematics for Industry, Springer Nature, Singapore, 2018, pp. 2738, DOI 10.1007/9789811062834_3 .
Abstract
This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed. 
A. Mielke, Uniform exponential decay for reactiondiffusion systems with complexbalanced massaction kinetics, in: Patterns of Dynamics, P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds., Proceedings in Mathematics & Statistics, Springer, 2017, pp. 149171, DOI 10.1007/9783319641737_10 .
Abstract
We consider reactiondiffusion systems on a bounded domain with noflux boundary conditions. All reactions are given by the massaction law and are assumed to satisfy the complexbalance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing.
We discuss three methods to obtain energydissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the logSobolev estimate and suitable handling of the reaction terms as well as the massconservation relations. The three methods are (i) a convexification argument based on the author's joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument of developed by Glitzky and Hünlich. 
M. Bulíček, A. Glitzky, M. Liero, Thermistor systems of p(x)Laplacetype with discontinuous exponents via entropy solutions, in: PDE 2015: Theory and Applications of Partial Differential Equations, H.Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., 10 of Discrete and Continuous Dynamical Systems, Series S, no. 4, American Institute of Mathematical Sciences, Springfield, 2017, pp. 697713.
Abstract
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the currentflow equation is of p(x)Laplaciantype with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L^{1} term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem. 
P. Colli, J. Sprekels, Optimal boundary control of a nonstandard CahnHilliard system with dynamic boundary condition and double obstacle inclusions, in: Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., 22 of Springer INdAM Series, Springer International Publishing AG, Cham, 2017, pp. 151182, DOI 10.1007/9783319644899 .
Abstract
In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P.PodioGuidugli in Ric. Mat. 55 (2006), pp.105118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the LaplaceBeltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 3558, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 130, for the case of (differentiable) logarithmic potentials and perform a socalled "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired firstorder necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials. 
N. Ahmed, A. Linke, Ch. Merdon, Towards pressurerobust mixed methods for the incompressible NavierStokes equations, in: Finite Volumes for Complex Applications VIII  Methods and Theoretical Aspects, FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 199 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing AG, Cham, 2017, pp. 351359.

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rateindependent damage in thermoviscoelastodynamics, in: MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1012009/20.
Abstract
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the ratedependent heat equation, and with the momentum balance featuring inertia and viscosity according to KelvinVoigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a timedependent Dirichlet loading. 
M. Thomas, E. Bonetti, E. Rocca, R. Rossi, A rateindependent gradient system in damage coupled with plasticity via structured strains, in: Gradient Flows: From Theory to Application, B. Düring, C.B. Schönlieb, M.Th. Wolfram, eds., 54 of ESAIM Proceedings and Surveys, EDP Sciences, 2016, pp. 5469.
Abstract
This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and RoyerCarfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rateindependent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers. 
A. Mielke, R. Rossi, G. Savaré, BalancedViscosity solutions for multirate systems, in: MURPHYSHSFS2014: 7th MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1012010/26.
Abstract
Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rateindependent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finitedimensional setting, and regularize both the static equation and the rateindependent flow rule by adding viscous dissipation terms with coefficients ε^{α} and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishingviscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rateindependent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1. 
A. Mielke, Deriving effective models for multiscale systems via evolutionary $Gamma$convergence, in: Control of SelfOrganizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235251.

A. Mielke, Evolutionary relaxation of a twophase model, in: Scales in Plasticity, MiniWorkshop, November 814, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 30273030.

A. Mielke, On thermodynamical couplings of quantum mechanics and macroscopic systems, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 331348.
Abstract
Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradientflow contribution, which satisfy a particular noninteraction condition:
We give three applications of the theory. First, we consider a finitedimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a onedimensional Schrödinger operator connected to a onedimensional heat equation on the left and on the right. Finally, we consider thermooptoelectronics, where the MaxwellBloch system of optics is coupled to the energydriftdiffusion system for semiconductor electronics. 
A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reactiondiffusion systems, in: Finite Volumes for Complex Applications VII  Methods and Theoretical Aspects  FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 77 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 275283.
Abstract
We prove a uniform Poincarelike estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reactiondiffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes. 
A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2  Mathematics for optoelectronic devices, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243256.

D. Knees, R. Kornhuber, Ch. Kraus, A. Mielke, J. Sprekels, C3  Phase transformation and separation in solids, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189203.

K. Götze, Free fall of a rigid body in a viscoelastic fluid, in: Geophysical Fluid Dynamics, Workshop, February 1822, 2013, 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 554556.

A. Mielke, Gradient structures and dissipation distances for reactiondiffusion systems, in: Material Theory, Workshop, Dezember 1620, 2013, A. Desimone, S. Luckhaus, L. Truskinovsky, eds., 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 34553458.

K. Götze, Maximal $L^p$regularity of a 2D fluidsolid interaction problem, in: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, W. Arendt, J.A. Ball, J. Behrndt, K.H. Förster, V. Mehrmann, C. Trunk, eds., 221 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2012, pp. 373384.
Abstract
We study a coupled system of equations which appears as a suitable linearisation of the model for the free motion of a rigid body in a Newtonian fluid in two space dimensions. For this problem, we show maximal Lpregularity estimates. 
A. Mielke, Multiscale gradient systems and their amplitude equations, in: Dynamics of Pattern, Workshop, Dezember 1622, 2012, 9 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2012, pp. 35883591.

R. HallerDintelmann, J. Rehberg, Maximal parabolic regularity for divergence operators on distribution spaces, in: Parabolic Problems: The Herbert Amann Festschrift, J. Escher, P. Guidotti, M. Hieber, P. Mucha, J.W. Pruess, Y. Shibata, G. Simonett, Ch. Walker, W. Zajaczkowski, eds., 80 of Progress in Nonlinear Differential Equations and Their Applications, Springer, Basel, 2011, pp. 313342.
Abstract
We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 
D. Knees, R. Rossi, C. Zanini, A vanishing viscosity approach in damage mechanics, in: Variational Methods for Evolution, Workshop, December 410, 2011, A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 31533155.

M. Thomas, Modeling and analysis of rateindependent damage and delamination processes, in: Proceedings of the 19th International Conference on Computer Methods in Mechanics (online only), 2011, pp. 16.

P. Krejčí, E. Rocca, J. Sprekels, Liquidsolid phase transitions in a deformable container, in: Continuous Media with Microstructure, B. Albers, ed., Springer, Berlin/Heidelberg, 2010, pp. 285300.

A. Mielke, Existence theory for finitestrain crystal plasticity with gradient regularization, in: IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, K. Hackl, ed., 21 of IUTAM Bookseries, Springer, Heidelberg, 2010, pp. 171183.

H. Gajewski, J.A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo, O. Minet, Image segmentation for the investigation of scatteredlight images when laseroptically diagnosing rheumatoid arthritis, in: Mathematics  Key Technology for the Future, W. Jäger, H.J. Krebs, eds., Springer, Heidelberg, 2008, pp. 149161.

A. Mielke, Numerical approximation techniques for rateindependent inelasticity, in: Proceedings of the IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, B.D. Reddy, ed., 11 of IUTAM Bookseries, Springer, 2008, pp. 5363.

A. Petrov, J.A.C. Martins, M.D.P. Monteiro Marques, Mathematical results on the stability of quasistatic paths of elasticplastic systems with hardening, in: Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific, Singapore, 2007, pp. 167182.

A. Petrov, Thermally driven phase transformation in shapememory alloys, in: Analysis and Numerics of RateIndependent Processes, Workshop, February 26  March 2, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 605607.

M. Eleuteri, Some P.D.E.s with hysteresis, in: Free Boundary Problems. Theory and Applications, I.N. Figueiredo, J.F. Rodrigues, L. Santos, eds., 154 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 2007, pp. 159168.

A. Glitzky, Energy models where the equations are defined on different domains, in: GAMM Annual Meeting 2006  Berlin, Special Issue (Vol. 6, Issue 1) of PAMM (Proceedings of Applied Mathematics and Mechanics), WileyVCH Verlag, Weinheim, 2006, pp. 629630.

R. Hünlich, G. Albinus, H. Gajewski, A. Glitzky, W. Röpke, J. Knopke, Modelling and simulation of power devices for highvoltage integrated circuits, in: Mathematics  Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 401412.

I.V. Skrypnik, H. Gajewski, On the uniqueness of solutions to nonlinear elliptic and parabolic problems (in Russian), in: Differ. Uravn. i Din. Sist., dedicated to the 80th anniversary of the Academician Evgenii Frolovich Mishchenko, Suzdal, 2000, 236 of Tr. Mat. Inst. Steklova, Moscow, Russia, 2002, pp. 318327.

R. Hünlich, A. Glitzky, On energy estimates for electrodiffusion equations arising in semiconductor technology, in: Partial differential equations. Theory and numerical solution, W. Jäger, J. Nečas, O. John, K. Najzar, eds., 406 of Chapman & Hall Research Notes in Mathematics, Chapman & Hall, Boca Raton, FL, 2000, pp. 158174.

H.Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of SchrödingerPoisson systems into the driftdiffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 13281333.
Preprints, Reports, Technical Reports

TH. Eiter, On the Stokestype resolvent problem associated with timeperiodic flow around a rotating obstacle, Preprint no. 2876, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2876 .
Abstract, PDF (265 kByte)
Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated timeperiodic linear problem. 
P.É. Druet, Maximal mixed parabolichyperbolic regularity for the full equations of multicomponent fluid dynamics, Preprint no. 2869, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2869 .
Abstract, PDF (561 kByte)
We consider a NavierStokesFickOnsagerFourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolichyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the shorttime existence of strong solutions for a typical initial boundaryvalueproblem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blowup or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volumeadditive mixtures. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Wellposedness and optimal control for a CahnHilliardOono system with control in the mass term, Preprint no. 2867, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2867 .
Abstract, PDF (358 kByte)
The paper treats the problem of optimal distributed control of a CahnHilliardOono system in R^{d}, 1 ≤ d ≤ 3 with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case d = 2. In the rest of the work, we study the necessary firstorder optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain 
P. Colli, A. Signori, J. Sprekels, Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory, Preprint no. 2863, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2863 .
Abstract, PDF (369 kByte)
A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a firstorder approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a nonconserved order parameter) and the socalled thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initialboundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fréchet differentiability between suitable Banach spaces is shown for the controltostate operator, and meaningful firstorder necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given. 
K. Hopf, Singularities in $L^1$supercritical FokkerPlanck equations: A qualitative analysis, Preprint no. 2860, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2860 .
Abstract, PDF (402 kByte)
A class of nonlinear FokkerPlanck equations with superlinear drift is investigated in the L^{1}supercritical regime, which exhibits a finite critical mass. The equations have a formal Wassersteinlike gradientflow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finitetime appearance constitutes a primary technical difficulty. This paper aims at a globalintime qualitative analysis  the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D KaniadakisQuarati model for BoseEinstein particles, and thus provides a first rigorous result on the continuation beyond blowup and longtime asymptotic behaviour for this model. 
M. Horák, M. Kružík, P. Pelech, A. Schlömerkemper, Gradient polyconvexity and modeling of shape memory alloys, Preprint no. 2851, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2851 .
Abstract, PDF (526 kByte)
We show existence of an energetic solution to a model of shape memory alloys in which the elastic energy is described by means of a gradientpolyconvex functional. This allows us to show existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Admissible deformations do not necessarily have integrable second derivatives. Under suitable assumptions, our model allows for solutions which are orientationpreserving and globally injective everywhere in the domain representing the specimen. Theoretical results are supported by threedimensional computational examples. This work is an extended version of [36]. 
P. Colli, A. Signori, J. Sprekels, Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities, Preprint no. 2845, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2845 .
Abstract, PDF (376 kByte)
This paper treats a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the $L^1$norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the socalled “deep quench approximation” in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, firstorder necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls. 
P. Krejčí, E. Rocca, J. Sprekels, Analysis of a tumor model as a multicomponent deformable porous medium, Preprint no. 2842, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2842 .
Abstract, PDF (270 kByte)
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phasedependent elasticity coefficients. The resulting PDE system couples two CahnHilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reactiondiffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initialboundary value problem has a solution in appropriate function spaces. 
R. Lasarzik, On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics, Preprint no. 2834, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2834 .
Abstract, PDF (264 kByte)
We define the concept of energyvariational solutions for the NavierStokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energyvariational solutions and thus weak solutions in any space dimension for the NavierStokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce wellposedness for these equations. 
X. Liu, M. Thomas, E. Titi, Wellposedness of Hibler's dynamical seaice model, Preprint no. 2833, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2833 .
Abstract, PDF (324 kByte)
This paper establishes the localintime wellposedness of solutions to an approximating system constructed by mildly regularizing the dynamical sea ice model of it W.D. Hibler, Journal of Physical Oceanography, 1979. Our choice of regularization has been carefully designed, prompted by physical considerations, to retain the original coupled hyperbolicparabolic character of Hibler's model. Various regularized versions of this model have been used widely for the numerical simulation of the circulation and thickness of the Arctic ice cover. However, due to the singularity in the ice rheology, the notion of solutions to the original model is unclear. Instead, an approximating system, which captures current numerical study, is proposed. The wellposedness theory of such a system provides a firststep groundwork in both numerical study and future analytical study. 
P. Colli, G. Gilardi, J. Sprekels, Wellposedness for a class of phasefield systems modeling prostate cancer growth with fractional operators and general nonlinearities, Preprint no. 2832, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2832 .
Abstract, PDF (310 kByte)
This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phasefield model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. bf 30 (2020), 12531295]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a AllenCahn equation with doublewell potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties. 
TH. Eiter, K. Hopf, A. Mielke, LerayHopf solutions to a viscoelastic fluid model with nonsmooth stressstrain relation, Preprint no. 2829, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2829 .
Abstract, PDF (342 kByte)
We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the ZarembaJaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of globalintime weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor. 
R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuationtheory perspective, Preprint no. 2826, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2826 .
Abstract, PDF (522 kByte)
Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the largedeviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode nondissipative effects. Our main contribution is an abstract framework, which for a given fluxdensity cost and a quasipotential, provides a decomposition into dissipative and nondissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems  independent copies of jump processes, zerorange processes, chemicalreaction networks in complex balance and latticegas models. 
D. Bothe, W. Dreyer, P.É. Druet, Multicomponent incompressible fluids  An asymptotic study, Preprint no. 2825, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2825 .
Abstract, PDF (519 kByte)
This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:(i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the nonappropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi or Gammaconvergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDEsystem relying on the equations of balance for partial masses, momentum and the internal energy.

A. Glitzky, M. Liero, G. Nika, A coarsegrained electrothermal model for organic semiconductor devices, Preprint no. 2822, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2822 .
Abstract, PDF (374 kByte)
We derive a coarsegrained model for the electrothermal interaction of organic semiconductors. The model combines stationary driftdiffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the GaussFermi integral and mobility functions depending on the temperature, chargecarrier density, and field strength, which is required for a proper description of organic devices. 
K. Hopf, Weakstrong uniqueness for energyreactiondiffusion systems, Preprint no. 2808, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2808 .
Abstract, PDF (444 kByte)
We establish weakstrong uniqueness and stability properties of renormalised solutions to a class of energyreactiondiffusion systems, which genuinely feature crossdiffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weakstrong uniqueness is obtained for general entropydissipating reactions without growth restrictions, and certain models with a nonintegrable diffusive flux. The results also apply to a class of (isoenergetic) reactioncrossdiffusion systems. 
J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, Preprint no. 2807, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2807 .
Abstract, PDF (489 kByte)
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where nonintegrable diffusion fluxes or reaction terms appear. 
M.H. Farshbaf Shaker, M. Thomas, Analysis of a compressible Stokesflow with degenerating and singular viscosity, Preprint no. 2786, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2786 .
Abstract, PDF (730 kByte)
In this paper we show the existence of a weak solution for a compressible singlephase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a densitydependent viscosity. The analysis is based on an implicit timediscrete scheme and a Galerkinapproximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of pLaplacian type in the transport equation that allows for refined compactness arguments on subdomains. 
A. Glitzky, M. Liero, G. Nika, An effective bulksurface thermistor model for largearea organic lightemitting diodes, Preprint no. 2757, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2757 .
Abstract, PDF (315 kByte)
The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of largearea organic lightemitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the threedimensional bulk glass substrate and two semilinear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the currentflow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixedpoint theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used. 
R. Lasarzik, Analysis of a thermodynamically consistent NavierStokesCahnHilliard model, Preprint no. 2739, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2739 .
Abstract, PDF (305 kByte)
In this paper, existence of generalized solutions to a thermodynamically consistent NavierStokesCahnHilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measurevalued and dissipative solutions. The measurevalued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measurevalued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weakstrong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions. 
P. Colli, G. Gilardi, J. Sprekels, Optimal control of a phase field system of Caginalp type with fractional operators, Preprint no. 2725, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2725 .
Abstract, PDF (360 kByte)
In their recent work “Wellposedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated controltostate operator is Fréchet differentiable between suitable Banach spaces, and meaningful firstorder necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables. 
L. Baňas, R. Lasarzik, A. Prohl, Numerical analysis for nematic electrolytes, Preprint no. 2717, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2717 .
Abstract, PDF (8587 kByte)
We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structureinheriting spacetime discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte. 
S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rateindependent damage models with gradient regularization, Preprint no. 2707, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2707 .
Abstract, PDF (3444 kByte)
This work provides a convergence analysis of a timediscrete scheme coupled with a finiteelement approximation in space for a model for partial, rateindependent damage featuring a gradient regularization as well as a nonsmooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rateindependent gradient damage is based on a Variable ADMMmethod to approximate the nonsmooth contribution. Spacediscretization is based on P1 finite elements and the algorithm directly couples the timestep size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BVfunctions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method. 
M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the FokkerPlanck operator, Preprint no. 2684, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2684 .
Abstract, PDF (2719 kByte)
We introduce a family of various finite volume discretization schemes for the FokkerPlanck operator, which are characterized by different weight functions on the edges. This family particularly includes the wellestablished ScharfetterGummel discretization as well as the recently developed squareroot approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly nontrivial while for large gradients the ScharfetterGummel scheme stands out compared to the others. 
A.F.M. TER Elst, A. Linke, J. Rehberg, On the numerical range of sectorial forms, Preprint no. 2667, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2667 .
Abstract, PDF (245 kByte)
We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a nonsymmetric secondorder elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H^{∞}angle for the H^{∞}calculus on L_{p} for all p ∈ (1, ∞) if the coefficients are real valued. 
R. Lasarzik, Maximal dissipative solutions for incompressible fluid dynamics, Preprint no. 2666, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2666 .
Abstract, PDF (225 kByte)
We introduce the new concept of maximal dissipative solutions for the NavierStokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages. 
D. Bothe, P.É. Druet, Mass transport in multicomponent compressible fluids: Local and global wellposedness in classes of strong solutions for general classone models, Preprint no. 2658, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2658 .
Abstract, PDF (509 kByte)
We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the FickOnsager or Maxwell Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the NavierStokes equations. The thermodynamic pressure is defined by the GibbsDuhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolichyperbolic type. We prove two theoretical results concerning the wellposedness of the model in classes of strong solutions: 1. The solution always exists and is unique for shorttimes and 2. If the initial data are sufficiently near to an equilibrium solution, the wellposedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity. 
H. Stephan, Millions of Perrin pseudoprimes including a few giants, Preprint no. 2657, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2657 .
Abstract, PDF (244 kByte)
The calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast algorithm but also an idea how most of them are structured to minimize the amount of numbers one have to test. We present a quick algorithm for testing Perrin pseudoprimes and develop some ideas on how Perrin pseudoprimes might be structured. This leads to some conjectures that still need to be proved.
We think that we have found well over 90% of all 20digit Perrin pseudoprimes. Overall, we have been able to calculate over 9 million Perrin pseudoprimes with our method, including some very large ones. The largest number found has 1436 digits. This seems to be a breakthrough, compared to the previously known just over 100,000 Perrin pseudoprimes, of which the largest have 20 digits.
In addition, we propose two sequences that do not provide any pseudoprimes up to 1,000,000,000 at all. 
A. Münch, B. Wagner, Selfconsistent field theory for a polymer brush. Part II: The effective chemical potential, Preprint no. 2649, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2649 .
Abstract, PDF (318 kByte)
The most successful meanfield model to describe the collective behaviour of the large class of macromolecular polymers is the selfconsistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the meanfield equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strongstretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching. 
A. Münch, B. Wagner, Selfconsistent field theory for a polymer brush. Part I: Asymptotic analysis in the strongstretching limit, Preprint no. 2648, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2648 .
Abstract, PDF (854 kByte)
In this study we consider the selfconsistent field theory for a dry, in compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the selfconsistent field theory in the strongstretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strongstretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the selfconsistent field theory. 
A. Glitzky, M. Liero, G. Nika, Analysis of a hybrid model for the electrothermal behavior of semiconductor heterostructures, Preprint no. 2636, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2636 .
Abstract, PDF (355 kByte)
We prove existence of a weak solution for a hybrid model for the electrothermal behavior of semiconductor heterostructures. This hybrid model combines an electrothermal model based on driftdiffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature. 
C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth CahnHilliard NavierStokes system with variable fluid densities, Preprint no. 2617, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2617 .
Abstract, PDF (11 MByte)
We are concerned with the simulation and control of a two phase flow model governed by a coupled CahnHilliard NavierStokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goaloriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (PODMOR) in order to replace highfidelity models by low order surrogates. In particular, we combine POD with spaceadapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property. 
A.F.M. TER Elst, H. Meinlschmidt, J. Rehberg, Essential boundedness for solutions of the Neumann problem on general domains, Preprint no. 2574, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2574 .
Abstract, PDF (220 kByte)
Let the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.
Talks, Poster

D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modeling and simulation of charge transport in perovskite solar cells, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021.

S. Schindler, Selfsimilar diffusive equilibration for a coupled reactiondiffusion system with massaction kinetics, SFB910: International Conference on Control of SelfOrganizing Nonlinear Systems (Hybrid Event), August 29  September 2, 2021, Technische Universität Berlin, Potsdam, September 1, 2021.

A. Alphonse, Directional differentiability and optimal control for elliptic quasivariational inequalities, Workshop ``Challenges in Optimization with Complex PDESystems'' (Online Workshop), February 14  20, 2021, Mathematisches Forschungsinstitut Oberwolfach, February 17, 2021.

A. Alphonse, Directional differentiability and optimal control for elliptic quasivariational inequalities (online talk), Meeting of the Scientific Advisory Board of WIAS, WIAS Berlin, March 12, 2021.

A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (coauthors: Michael Hintermüller and Carlos Rautenberg, online talk), 91th Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Session DFGPP 1962 Nonsmooth and Complementaritybased Distributed Parameter Systems, March 15  19, 2021, Universität Kassel, March 16, 2021.

A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (online talk), Annual Meeting of the DFG SPP 1962 (Virtual Conference), March 24  25, 2021, WIAS Berlin, March 25, 2021.

X. Liu, Justification of the primitive equations (online talk), Global Scientist Interdisciplinary online Forum 2021, Southern University of Science and Technology, Shenzhen, China, January 9, 2021.

G. Nika, Derivation of an effective bulksurface thermistor model for OLEDs (online presentation), WIAS Berlin (Online Event), September 6  9, 2021.

P. Pelech, Balancedviscosity solution for a PenroseFife model with rateindependent friction (online talk), Jahrestreffen des SPP 2256 (Online Event), September 22  24, 2021, Universität Regensburg, September 22, 2021.

P. Pelech, On compatibility of the natural configuration framework with general equation of nonequilibrium reversibleirreversible coupling (GENERIC), 16thJoint European Thermodynamics Conference (Hybrid Event), June 14  18, 2021, Charles University Prague, Czech Republic, June 14, 2021.

P. Pelech, Separately global solutions to rateindependent systems  applications to largestrain deformations of damageable solids (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 19, 2021.

A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reactiondiffusion systems (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 17, 2021.

P. Vágner , Generalized NernstPlanckPoisson model of solid oxide YSZ LSM O_2 electrode interface (online talk), 72nd Annual International Society of Electrochemistry (ISE) Meeting (Online Event), August 29  September 3, 2021, Seoul National University, Jeju Island, Korea (Republic of), September 3, 2021.

P. Vágner , Generalized NernstPlanckPoisson model of solid oxide YSZ LSM O_2 electrode interface (online talk), AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021, WIAS Berlin, September 7, 2021.

B. Gaudeul, J. Fuhrmann, Two entropic finite volume schemes for a NernstPlanckPoisson system with ion volume constraints, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021.

A. Glitzky, A coarsegrained electrothermal model for organic semiconductor devices (online talk), DMVÖMG Jahrestagung 2021 (Online Event), September 27  October 1, 2021, Universität Passau, September 29, 2021.

M. Thomas, Convergence analysis for fully discretized damage and phasefield fracture models (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13  17, 2021, WIAS Berlin, September 14, 2021.

G. Dong, A class of secondorder quasilinear PDEs and their applications, 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (FBP 2021, Online Event), September 13  17, 2021, WIAS, Berlin, September 16, 2021.

P.E. Druet, Modeling and analysis for multicomponent incompressible fluids (online talk), 8th European Congress of Mathematics, Minisymposium ID 51 ``Partial differential equations describing farfromequilibrium open systems'' (Online Event), June 20  26, 2021, European Mathematical Society, June 23, 2021.

P.E. Druet, Wellposedness results for mixedtype systems modelling pressuredriven multicomponent fluid flows (online talk), 8th European Congress of Mathematics, Minisymposium ID 42 ``Multicomponent diffusion in porous media'' (Online Event), June 20  26, 2021, European Mathematical Society, June 22, 2021.

TH. Eiter, Viscoelastic fluid flows with nonsmooth stress dissipation (online talk), DMVÖMG Jahrestagung 2021 (Online Event), September 27  October 1, 2021, Universität Passau, September 29, 2021.

P. Farrell, Y. Hadjimichael, Ch. Merdon, T. Streckenbach, Toward charge transport in bent nanowires, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021.

M. Liero, Machine learning and PDEs with Julia (online talk), FUHRI2021: Finite Volume Methoods for Realworld AppIications (Online Event), April 29, 2021, WIAS Berlin, April 29, 2021.

M. Liero, Heat and carrier flow in organic semiconductor devices  modeling, analysis, and simulation (online talk), AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021, WIAS Berlin, September 6, 2021.

A. Mielke, Effective models for materials and interfaces with multiple scales (online poster session), CRC 1114 Conference 2021 (online participation), March 1  3, 2021.

A. Mielke, Gradient structures and EDP convergence for reaction and diffusion (online talk), Recent Advances in Gradient Flows, Kinetic Theory, and ReactionDiffusion Equations (Online Event), July 13  16, 2021, Universität Wien, July 15, 2021.

A. Mielke, Thermoviscoelasticity at finite strain (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 19, 2021.

M. Heida, Stochastic homogenization in randomly perforated domains (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30  October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 2, 2020.

M. Heida, Stochastic homogenization on perforated domains, FriedrichAlexander Universität ErlangenNürnberg, Department Mathematik, July 2, 2020.

M. Heida, Stochastic homogenization on perforated domains (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23  25, 2020, WIAS Berlin, November 24, 2020.

D.R.M. Renger, Fast reaction limits via Γconvergence of the Flux Rate Functional, Variational Methods for Evolution, September 13  19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

A. Stephan, On gradient flows and gradient systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 11, 2020.

A. Stephan, On gradient systems and applications to interacting particle systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 25, 2020.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

A. Glitzky, A hybrid model for the electrothermal behaviour of semiconductor devices (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30  October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

K. Hopf, Global existence analysis of energyreactiondiffusion systems, Workshop ``Variational Methods for Evolution'', September 13  19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

M. Thomas, Modeling and analysis of flows of concentrated suspensions (online talk), Colloquium of the RTG 2339 ``Interfaces, Complex Structures, and Singular Limits'' (Online Event), Universität Regensburg, July 10, 2020.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Thomas, Thermodynamical modelling via energy and entropy functionals (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin.

M. Thomas, WeierstraßGruppe "VolumenGrenzschichtProzesse", Sitzung des Wissenschaftlichen Beirats, WIAS Berlin, September 18, 2020.

TH. Eiter, Spatially asymptotic structure of timeperiodic NavierStokes flows (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23  25, 2020, WIAS Berlin, November 24, 2020.

TH. Eiter , Spatially asymptotic structure of timeperiodic NavierStokes flows (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30  October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

R. Lasarzik, Dissipative solutions in the context of the numerical approximation of nematic electrolytes (online talk), Oberseminar Numerik, Universität Bielefeld, Fakultät für Mathematik, June 23, 2020.

M. Liero, A. Mielke, Analysis for thermomechanical models with internal variables, Presentation of project proposals in DFG SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Liero, Evolutionary Gammaconvergence for multiscale problems (online talks), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin, October 15, 2020.

A. Mielke, Finitestrain viscoelasticity with temperature coupling, Calculus of Variations and Applications, January 27  February 1, 2020, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, January 28, 2020.

A. Mielke, Gradient systems and evolutionary Gammaconvergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'' (Online Event), JuliusMaximiliansUniversität Würzburg, June 5, 2020.

A. Mielke, On finitestrain thermoviscoelasticity, Mechanics of Materials: Towards Predictive Methods for Kinetics in Plasticity, Fracture, and Damage, March 8  14, 2020, Mathematisches Forschungszentrum Oberwolfach, March 12, 2020.

A. Mielke, Differential equations as gradient flows, with applications in mechanics, stochastics, and chemistry (online talk), Würzburger Mathematisches Kolloquium (Online Event), JuliusMaximiliansUniversität Würzburg, November 9, 2020.

A. Mielke, EDPconvergence for multiscale gradient systems with applications to fastslow reaction systems (online talk), One World Dynamics Seminar (Online Event), Technische Universität München, November 13, 2020.

A. Mielke, Global existence for finitestrain viscoelasticity with temperature coupling (online talk), One World Dynamics Seminar (Online Event), University of Bath, UK, December 1, 2020.

A. Mielke, Similarity solutions for Kolmogorov's twoequation model for turbulence, Workshop on Control of SelfOrganizing Nonlinear Systems, September 2  3, 2020, CRC 910, Technische Universität Berlin, September 2, 2020.

A. Mielke, Variational structures for the analysis of PDE systems (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin, October 13, 2020.

J. Rehberg, Explicit and uniform estimates for second order divergence operators on $L^p$ spaces, Oberseminar ``Analysis und Theoretische Physik'', Leibniz Universität Hannover, Institut für Angewandte Mathematik, January 28, 2020.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, Visit of the Scientific Advisory Board of MATH+, November 11, 2019.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, 1st MATH+ Day, Berlin, December 13, 2019.

M. Heida, A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, Zeuthen, May 20  22, 2019.

M. Heida, Convergences of the squareroot approximation scheme to the FokkerPlanck operator, ``9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019), July 15  19, 2019, Universitat de València, Spain, July 17, 2019.

M. Heida, The SQRA operator: Convergence behaviour and applications, Universität Wien, Fakultät für Mathematik, Lehrstuhl Analysis, Austria, March 19, 2019.

M. Heida, The fractional pLaplacian emerging from discrete homogenization of the random conductance model with degenerate ergodic weights, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Universität Wien, Technische Universität Wien, Austria, February 19, 2019.

M. Heida, What is... SQRA discretization of the FokkerPlanck equation?, CRC1114 Colloquium, Freie Universität Berlin, SFB 1114, April 25, 2019.

G. Nika, An existence result for a class of electrothermal driftdiffusion models with GaussFermi statistics for organic semiconductors, DMVJahrestagung 2019, September 23  26, 2019, KIT  Karlsruher Institut für Technologie.

D. Peschka, Dynamic contact angles via gradient flows, 694. WEHeraeusSeminar ``Wetting on Soft or Microstructured Surfaces'', Bad Honnef, April 10  13, 2019.

A. Stephan, Evolutionary Gammaconvergence for a linear reactiondiffusion system with different time scales, COPDESCWorkshop ``Calculus of Variation and Nonlinear Partial Differential Equations", March 25  28, 2019, Universität Regensburg, March 26, 2019.

A. Stephan, On evolution semigroups and Trotter product operatornorm estimates, Operator Theory and Krein Spaces, December 19  22, 2019, Technische Universität Wien, Austria, December 20, 2019.

A. Zafferi, Some regularity results for a nonisothermal CahnHilliard model, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, 12th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis, December 4  6, 2019, University of Oxford, Mathematical Institute, UK, December 6, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, Seminar Forschergruppe 2402: Research Unit  Rough paths, stochastic partial differential equations and related topics, Technische Universität Berlin, Institut für Mathematik, December 12, 2019.

W. van Zuijlen, Minicourse on Besov spaces IIII, Junior Trimester Program: Randomness, PDEs and Nonlinear Fluctuations (Sept. 2 to Dec. 19, 2019), October 16  November 6, 2019, Hausdorff Research Institute for Mathematics (HIM), Bonn.

A. Glitzky, An existence result for a class of electrothermal driftdiffusion models with GaussFermi statistics for organic semiconductors, ``Partial Differential Equations in Fluids and Solids" (PDE2019), September 9  13, 2019, WIAS Berlin, September 12, 2019.

A. Glitzky, Driftdiffusion problems with GaussFermi statistics and fielddependent mobility for organic semiconductor devices, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Universität Wien, Technische Universität Wien, Austria, February 22, 2019.

K. Hopf, On the singularity formation and relaxation to equilibrium in 1D FokkerPlanck model with superlinear drift, Winter School ``Gradient Flows and Variational Methods in PDEs'', November 25  29, 2019, Universität Ulm, November 25, 2019.

K. Hopf, On the singularity formation and relaxation to equilibrium in 1D FokkerPlanck model with superlinear drift, Gradient Flows and Variational Methods in PDEs, November 25  29, 2019, Universität Ulm, November 25, 2019.

M. Thomas, Analysis for the discrete approximation of gradientregularized damage models, Mathematics Seminar Brescia, Università degli Studi di Brescia, Italy, March 13, 2019.

M. Thomas, Analysis for the discrete approximation of gradientregularized damage models, PDE Afternoon, Universität Wien, Austria, April 10, 2019.

M. Thomas, Analytical and numerical aspects for the approximation of gradientregularized damage models, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS A3226 ``PhaseField Models in Simulation and Optimization'', July 15  19, 2019, Valencia, Spain, July 17, 2019.

M. Thomas, Analytical and numerical aspects of rateindependent gradientregularized damage models, Conference ``Dynamics, Equations and Applications (DEA 2019)'', Session D444 ``Topics in the Mathematical Modelling of Solids'', September 16  20, 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems, MURPHYSHSFS 2019 Summer School on MultiRate Processes, SlowFast Systems and Hysteresis, June 17  19, 2019, Politecnico di Torino, Turin, Italy.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Mathematics for Mechanics, October 29  November 1, 2019, Czech Academy of Sciences, Prague, Czech Republic, October 31, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.

M. Thomas, Gradient structures for flows of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME775 ``Recent Advances in Understanding Suspensions and Granular Media Flow'', July 15  19, 2019, Valencia, Spain, July 17, 2019.

M. Thomas, Rateindependent evolution of sets and application to fracture processes, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

B. Wagner, Free boundary problems of active and driven hydrogels, PIMSGermany Workshop on Modelling, Analysis and Numerical Analysis of PDEs for Applications, June 24  26, 2019, Universität Heidelberg, Interdisciplinary Center for Scientific Computing and BIOQUANT Center, June 24, 2019.

M. Heida, The SQRA operator: Convergence behaviour and applications, Politechnico di Milano, Dipartimento di Matematica, Italy, March 13, 2019.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Colloquium of the Mathematical Institute, University of Oxford, UK, June 7, 2019.

R. Lasarzik, Weak entropic solutions to a model in induction hardening: Existence and weakstrong uniqueness, Decima Giornata di Studio Università di Pavia  Politecnico di Milano Equazioni Differenziali e Calcolo delle Variazioni, Politecnico di Milano, Italy, February 21, 2019.

M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Universität Wien, Technische Universität Wien, Austria, February 20, 2019.

A. Mielke, An existence result for thermoviscoelasticity at finite strains, Mathematics for Mechanics, October 29  November 1, 2019, Czech Academy of Sciences, Institute for Information Theory and Automation, Prague, Czech Republic, November 1, 2019.

A. Mielke, Effective kinetic relations and EDP convergence, COPDESCWorkshop ``Calculus of Variation and Nonlinear Partial Differential Equations'', March 25  28, 2019, Universität Regensburg, March 28, 2019.

A. Mielke, Effective kinetic relations and EDP convergence for gradient systems, Necas Seminar on Continuum Mechanics, Charles University, Prague, Czech Republic, March 18, 2019.

A. Mielke, Evolutionary Gammaconvergence for gradient systems, Mathematisches Kolloquium, AlbertLudwigsUniversität Freiburg, January 24, 2019.

A. Mielke, Gamma convergence of dissipation functionals and EDP convergence for gradient systems, 6th Applied Mathematics Symposium Münster: Recent Advances in the Calculus of Variations, September 16  19, 2019, Westfälische WilhelmsUniversität Münster, September 17, 2019.

A. Mielke, Gradient systems and evolutionary Gammaconvergence, DMVJahrestagung 2019, September 23  26, 2019, KIT  Karlsruher Institut für Technologie, September 24, 2019.

A. Mielke, On Kolmogorov's twoequation model for turbulence, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Universität Wien, Technische Universität Wien, Austria, February 21, 2019.

A. Mielke, On initialboundary value problems for materials with internal variables or temperature dependence, Workshop on Mathematical Methods in Continuum Physics and Engineering: Theory, Models, Simulation, November 6  7, 2019, Technische Universität Darmstadt, November 6, 2019.

A. Mielke, Pattern formation in coupled parabolic systems on extended domains, Fundamentals and Methods of Design and Control of Complex Systems  Introductory Lectures 2019/20 of CRC 910, Technische Universität Berlin, November 25, 2019.

A. Mielke, Variational methods in timedependent material models with finitestrain deformations, Hausdorff School on Modeling and Analysis of Evolutionary Problems in Materials Science, September 23  27, 2019, Hausdorff Center for Mathematics, Universität Bonn.

A. Mielke, EDP convergence for the membrane limit in the porous medium equation, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME13 9 ``Entropy Methods for Multidimensional Systems in Mechanics'', July 15  19, 2019, Valencia, Spain, July 19, 2019.

A. Mielke, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center  CRC 1114, May 20  22, 2019, SFB 1114, Freie Universität Berlin, Zeuthen, May 21, 2019.

A. Mielke, Gradient systems and the derivation of effective kinetic relations via EDP convergence, Material Theories, Statistical Mechanics, and Geometric Analysis: A Conference in Honor of Stephan Luckhaus' 66th Birthday, June 3  6, 2019, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, June 5, 2019.

A. Mielke, Transport versus growth and decay: The (spherical) HellingerKantorovich distance between arbitrary measures, Optimal Transport: From Geometry to Numerics, May 13  17, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Universität Wien, Austria, May 17, 2019.

J. Rehberg, An extrapolation for the LaxMilgram isomorphism for second order divergence operators, Oberseminar ``Angewandte Analysis'', Technische Universität Darmstadt, February 7, 2019.

J. Rehberg, Explicit and uniform estimates for second order divergence operators on $L^P$ spaces, Evolution Equations: Applied and Abstract Perspectives, October 28  November 1, 2019, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, October 31, 2019.

J. Rehberg, Maximal parabolic regularity for the treatment of real world problems, 12th Workshop on Analysis and Advanced Numerical Methods for Partial Differential Equations (not only) for Junior Scientists (AANMPDE 12), July 1  5, 2019, Österreichische Akademie der Wissenschaften, St. Wolfgang / Strobl, Austria, July 2, 2019.

J. Rehberg, Wellposedness for the KellerSegel model  based on a pioneering result of Herbert Amann, International Conference ``Nonlinear Analysis'' in Honor of Herbert Amann's 80th Birthday, June 11  14, 2019, Scuola Normale Superiore di Pisa, Cortona, Italy, June 11, 2019.

S. Reichelt, Pulses in FitzHughNagumo systems with rapidly oscillating coefficients, SIAM Annual Meeting, Minisymposium 101 ``Multiscale Analysis and Simulation on Heterogeneous Media'', July 9  13, 2018, Society for Industrial and Applied Mathematics, Oregon Convention Center (OCC), Portland, USA, July 12, 2018.

N. Rotundo, On a thermodynamically consistent coupling of quantum system and device equations, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium ``Mathematical Modeling of Charge Transport in Graphene and Low Dimensional Structures'', August 18  June 22, 2018, Budapest, Hungary, June 19, 2018.

M. Heida, Mathematische Mehrskalenmethoden in Natur und Technik, Seminar ``Angewandte Analysis'', Universität Konstanz, Institut für Mathematik, October 31, 2018.

M. Heida, On Gconvergence and stochastic twoscale convergences of the square root approximation scheme to the FokkerPlanck operator, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section S14 ``Applied Analysis'', March 19  23, 2018, Technische Universität München, March 21, 2018.

M. Heida, On convergence of the square root approximation scheme to the FokkerPlanck operator, Technische Universität Berlin, Institut für Mathematik, May 14, 2018.

M. Heida, On convergence of the square root approximation scheme to the FokkerPlanck operator, Oberseminar ``Optimierung'', HumboldtUniversität zu Berlin, Institut für Mathematik, May 29, 2018.

M. Heida, On convergences of the square root approximation scheme to the FokkerPlanck operator, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', February 11  17, 2018, Mathematisches Forschungsinstitut Oberwolfach, February 13, 2018.

A. Zafferi, Regularity results for a thermodynamically consistent nonisothermal CahnHilliard model, Summer School ``Dissipative Dynamical Systems and Applications'', September 3  7, 2018, University of Modena, Department of Physics, Informatics and Mathematics, Italy, September 6, 2018.

J. Sprekels, CahnHilliard systems with general fractional operators, Workshop ``Challenges in Optimal Control of Nonlinear PDESystems'', April 9  13, 2018, Mathematisches Forschungsinstitut Oberwolfach, April 9, 2018.

J. Sprekels, CahnHilliard systems with general fractionalorder operators, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 22, 2018.

M. Thomas, D. Peschka, B. Wagner, V. Mehrmann, M. Rosenau, Modeling and analysis of suspension flows, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section DFG Priority Programmes PP1748 ``Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretization Methods, Mechanical and Mathematical Analysis'', March 19  23, 2018, Technische Universität München, March 20, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.

M. Thomas, Analysis for the discrete approximation of damage and fracture, Applied Analysis Day, June 28  29, 2018, Technische Universität Dresden, Chair of Partial Differential Equations, June 29, 2018.

M. Thomas, Analysis for the discrete approximation of gradientregularized damage models, Workshop ``Women in Mathematical Materials Science'', November 5  6, 2018, Universität Regensburg, Fakultät für Mathematik, November 6, 2018.

M. Thomas, Analytical and numerical approach to a class of damage models, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 75 ``Mathematics and Materials: Models and Applications'', July 5  9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 6, 2018.

M. Thomas, Analytical and numerical aspects of damage models, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', November 29  30, 2018, Universität Würzburg, Institut für Mathematik, November 30, 2018.

M. Thomas, Gradient structures for flows of concentrated suspensions, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 18 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations and Related Fields'', July 5  9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 7, 2018.

M. Thomas, Optimization of the radiative emission for mechanically strained optoelectronic semiconductor devices, 9th International Conference ``Inverse Problems: Modeling and Simulation'' (IPMS 2018), Minisymposium M16 ``Inverse and Control Problems in Mechanics'', May 21  25, 2018, The Eurasian Association on Inverse Problems, Malta, May 24, 2018.

M. Thomas, Rateindependent evolution of sets & applications to damage and delamination, PDEs Friends, June 21  22, 2018, Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy, June 22, 2018.

M. Thomas, Reliable error estimates for phasefield models of brittle fracture, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

R. Lasarzik, Generalised solutions to the EricksenLeslie model describing liquid crystal flow, Università degli Studi di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, June 7, 2018.

A. Mielke, Construction of effective gradient systems via EDP convergence, Workshop on Mathematical Aspects of NonEquilibrium Thermodynamics, March 5  7, 2018, RheinischWestfälische Technische Hochschule Aachen, March 6, 2018.

A. Mielke, Energy, dissipation, and evolutionary Gamma convergence for gradient systems, Kolloquium ``Applied Analysis'', Universität Bremen, December 18, 2018.

S. Reichelt, Corrector estimates for elliptic and parabolic equations with periodic coefficients, Analysis Seminar, FriedrichAlexanderUniversität ErlangenNürnberg, Institut für Angewandte Mathematik, Erlangen, May 18, 2017.

S. Reichelt, Corrector estimates for elliptic and parabolic equations with periodic coefficients, Analysis Seminar, Universität Augsburg, Institut für Mathematik, May 23, 2017.

S. Reichelt, Corrector estimates for imperfect transmission problems, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

S. Reichelt, Pulses in FitzHughNagumo systems with periodic coefficients, Seminar ``Dynamical Systems and Applications'', Technische Universität Berlin, Institut für Mathematik, May 3, 2017.

A. Alphonse, A coupled bulksurface reactiondiffusion system on a moving domain, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 23  28, 2017, Mathematisches Forschungszentrum Oberwolfach, January 25, 2017.

A. Alphonse, Optimal control of elliptic and parabolic quasivariational inequalities, Annual Meeting of the DFG Priority Programme 1962, October 9  11, 2017, Kremmen (Sommerfeld), October 10, 2017.

M. Heida, On Gconvergence and stochastic twoscale convergences of the squareroot approximation scheme to the FokkerPlanck operator, GAMMWorkshop on Analysis of Partial Differential Equations, September 27  29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

M. Liero, On entropytransport problems and the HellingerKantorovich distance, Seminar of Team EDPAIRSEACVGI, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble, France, January 26, 2017.

D.R.M. Renger, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27  March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, March 1, 2017.

J. Sprekels, A nonstandard viscous CahnHilliard system with dynamic boundary condition and the DCH, Analysis of Boundary Value Problems for PDEs  Workshop on the Occasion of the 70th Birthday of Gianni Gilardi, Pavia, Italy, February 20, 2017.

J. Sprekels, Optimal control of PDEs: From basic principles to hard applications, International School ``Frontiers in Partial Differential Equations and Solvers'', May 22  25, 2017, University of Pavia, Department of Mathematics, Italy.

J. Sprekels, Wellposedness and optimal control of a nonstandard CahnHilliard system with dynamic boundary condition, Fudan University, School of Mathematical Sciences, China, April 10, 2017.

N. Ahmed, A numerical study of residual based variational multiscale methods for turbulent incompressible flow problems, American University of the Middle East, Dasman, Kuwait, November 2, 2017.

P.É. Druet, Analysis of recent NernstPlanckPoissonNavierStokes systems of electrolytes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

P.É. Druet, Existence of weak solutions for improved NernstPlanckPoisson models of compressible electrolytes, Seminar EDE, Czech Academy of Sciences, Institute of Mathematics, Department of Evolution Differential Equations (EDE), Prague, Czech Republic, January 10, 2017.

M. Hintermüller, Generalized Nash games with partial differential equations, Kolloquium Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, June 20, 2017.

TH. Koprucki, Mathematical knowledge management as a route to sustainability in mathematical modeling and simulation, 2nd Leibniz MMS Days 2017, February 22  24, 2017, Technische Informationsbibliothek (TIB), Hannover, February 22, 2017, DOI 10.5446/21908 .

A. Mielke, A geometric approach to reactiondiffusion equations, Institutskolloquium, Universität Potsdam, Institut für Mathematik, January 25, 2017.

A. Mielke, Global existence results for viscoplasticity, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 10, 2017.

A. Mielke, Optimal transport versus growth and decay, International Conference ``Calculus of Variations and Optimal Transportation'' in the Honor of Yann Brenier for his 60th Birthday, January 9  11, 2017, Institut Henri Poincaré, Paris, France, January 11, 2017.

A. Mielke, Perspectives for gradient flows, GAMMWorkshop on Analysis of Partial Differential Equations, September 27  29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

A. Mielke, Uniform exponential decay for energyreactiondiffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italy, March 21, 2017.

J. Rehberg, Explicit and uniform resolvent estimates for second order divergence operators on $L^p$ spaces, Oberseminar Analysis, Technische Universität Darmstadt, Fachbereich Mathematik, November 9, 2017.

J. Rehberg, On optimal elliptic Sobolev regularity, Oberseminar Prof. Ira Neitzel, Rheinische FriedrichWilhelmsUniversität Bonn, Institut für Numerische Simulation, February 2, 2017.

E. Cinti, Quantitative flatness results and BVestimates for nonlocal minimal surfaces, Seminario di Equazioni Differenziali, Università degli Studi di Roma ``Tor Vergata'', Italy, April 19, 2016.

K. Disser, Convergence for gradient systems of slow and fast chemical reactions, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, Braunschweig, March 11, 2016.

K. Disser, Econvergence to the quasisteadystate approximation in systems of chemical reactions, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22  26, 2016, WIAS Berlin, Berlin, February 25, 2016.

S. Reichelt, Error estimates for elliptic and parabolic equations with oscillating coefficients, Karlstad Applied Analysis Seminar, Karlstad University, Department of Mathematics and Computer Science, Sweden, April 13, 2016.

S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary $Gamma$convergence, Joint Annual Meeting of DMV and GAMM, Young Researchers' Minisymposium ``Multiscale Evolutionary Problems'', March 7  11, 2016, Technische Universität Braunschweig, March 7, 2016.

S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary Gammaconvergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25  29, 2016.

S. Reichelt, Homogenization of CahnHilliardtype equations via gradient structures, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 2 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equation'', July 1  5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 3, 2016.

TH. Frenzel, EDPconvergence for delamination and a wiggly energy model, 2nd Berlin Dresden Prague Würzburg Workshop on Mathematics of Continuum Mechanics, Technische Universität Dresden, Fachbereich Mathematik, Dresden, December 5, 2016.

TH. Frenzel, Evolutionary Gammaconvergence for a delamination model, Workshop on Industrial and Applied Mathematics 2016, 5th Symposium of German SIAM Student Chapters, August 31  September 2, 2016, University of Hamburg, Department of Mathematics, Hamburg, September 1, 2016.

TH. Frenzel, Evolutionary Gammaconvergence for amplitude equations and for wiggly energy models, Winter School 2016: Calculus of Variations in Physics and Materials Science, Würzburg, February 15  19, 2016.

TH. Frenzel, Examples of evolutionary Gammaconvergence, Workshop on Industrial and Applied Mathematics 2016, 5th Symposium of German SIAM Student Chapters, Hamburg, August 31  September 2, 2016.

M. Heida, Large deviation principle for a stochastic AllenCahn equation, 9th European Conference on Elliptic and Parabolic Problems, May 23  27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 25, 2016.

M. Liero, Gradient structures for reactiondiffusion systems and optimal entropytransport problems, Workshop ``Variational and Hamiltonian Structures: Models and Methods'', July 11  15, 2016, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, July 11, 2016.

M. Liero, OLEDs  a hot matter? Electrothermal modeling of OLEDs., sc Matheon Workshop, 9th Annual Meeting ``Photonic Devices'', March 3  4, 2016, Zuse Institute Berlin, Berlin, March 4, 2016.

M. Liero, On $p(x)$Laplace thermistor models describing eletrothermal feedback in organic semiconductors, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 23 ``Charge Transport in Semiconductor Materials: Emerging and Established Mathematical Topics'', June 13  17, 2016, Universidade de Santiago de Compostela, Spain, June 15, 2016.

M. Liero, On $p(x)$Laplace thermistor models describing eletrothermal feedback in organic semiconductors, Joint Annual Meeting of DMV and GAMM, Section 14 ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, Braunschweig, March 9, 2016.

M. Liero, On electrothermal feedback in organic lightemitting diodes, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

M. Liero, On geodesic curves and convexity of functionals with respect to the HellingerKantorovich distance, Workshop ``Optimal Transport and Applications'', November 7  11, 2016, Scuola Normale Superiore, Dipartimento di Matematica, Pisa, Italy, November 10, 2016.

E. Valdinoci, A notion of fractional perimeter and nonlocal minimal surfaces, Seminar, Universitá del Salento, Dipartimento di Matematics e Fisica ``Ennio de Giorgi'', Lecce, Italy, June 22, 2016.

E. Valdinoci, Interior and boundary properties of nonlocal minimal surfaces, Calcul des Variations & EDP, Université AixMarseille, Institut de Mathématiques de Marseille, France, February 25, 2016.

E. Valdinoci, Interior and boundary properties on nonlocal minimal surfaces, 3rd Conference on Nonlocal Operators and Partial Differential Equations, June 27  July 1, 2016, Bedlewo, Poland, June 27, 2016.

E. Valdinoci, Nonlocal Equations and Applications, Spring School on Nonlinear PDEs and Related Problems, January 15  19, 2016, African Institute for Mathematical Sciences (AIMS), Mbour, Senegal.

E. Valdinoci, Nonlocal equations and applications, Calculus of Variations and Nonlinear Partial Differential Equations, May 16  27, 2016, Columbia University, Department of Mathematics, New York, USA, May 25, 2016.

E. Valdinoci, Nonlocal equations from different points of view, Research Seminar, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, August 31, 2016.

E. Valdinoci, Nonlocal equations from various perspectives, PIMS Workshop on Nonlocal Variational Problems and PDEs, June 13  17, 2016, University of British Columbia, Vancouver, Canada, June 13, 2016.

E. Valdinoci, Nonlocal minimal surfaces, Analysis, PDEs, and Geometry Seminar, Monash University, Clayton, Australia, August 9, 2016.

E. Valdinoci, Nonlocal minimal surfaces and phase transitions, Seminar, University of Leeds, School of Mathematics, UK, April 19, 2016.

E. Valdinoci, Nonlocal minimal surfaces, a geometric and analytic insight, Seminar on Differential Geometry and Analysis, OttovonGuerickeUniversität Magdeburg, January 18, 2016.

E. Valdinoci, Nonlocal minimal surfaces: Regularity and quantitative properties, Conference on Recent Trends on Elliptic Nonlocal Equations, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, June 9, 2016.

E. Valdinoci, Planelike solutions in periodic media, Asymptotic Patterns in Variational Problems: PDE and Geometric Aspects, September 25  30, 2016, Banff International Research Station for Mathematical Innovation and Discovery, Oaxaca, Mexico, September 28, 2016.

E. Valdinoci, Superfici minime e transizioni di fase (non)locali, EDP e Dintorni II Incontro, Università di Bari, Dipartimento di Matematica, Italy, May 13, 2016.

M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, M. Bär, A. Mielke, Competing patterns in antisymmetrically coupled SwiftHohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of SelfOrganizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4  8, 2016.

A. Glitzky, $p(x)$Laplace thermistor models for electrothermal effects in organic semiconductor devices, 7th European Congress of Mathematics (7ECM), Minisymposium 22 ``Mathematical Methods for Semiconductors'', July 18  22, 2016, Technische Universität Berlin, July 22, 2016.

A. Glitzky, $p(x)$Laplace thermistor models for electrothermal feedback in organic semiconductor devices, 9th European Conference on Elliptic and Parabolic Problems, May 23  27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

M. Thomas, Analysis and optimization for edgeemitting semiconductor heterostructures, 7th European Congress of Mathematics (ECM), session CS8A, July 18  22, 2016, Technische Universität Berlin, Berlin, July 19, 2016.

M. Thomas, Analysis and optimization for edgeemitting semiconductor heterostructures, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 2 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equation'', July 1  5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 3, 2016.

M. Thomas, Coupling rateindependent and ratedependent processes: Delamination models in viscoelastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, 7th European Congress of Mathematics (ECM), minisymposium ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18  22, 2016, Technische Universität Berlin, Berlin, July 22, 2016.

M. Thomas, Energetic concepts for coupled rateindependent and ratedependent processes: Damage & delamination in viscoelastodynamics, International Conference ``Mathematical Analysis of Continuum Mechanics and Industrial Applications II'' (CoMFoS16), October 22  24, 2016, Kyushu University, Fukuoka, Japan.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, special session ``Ratedependent and Rateindependent Evolution Problems in Continuum Mechanics: Analytical and Numerical Aspects'', July 1  5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 4, 2016.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22  26, 2016, WIAS Berlin, Berlin, February 26, 2016.

M. Thomas, Nonsmooth PDEs in material failure: Towards dynamic fracture, Joint Annual Meeting of DMV and GAMM, Section 14 ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, March 10, 2016.

M. Thomas, Rateindependent evolution of sets, INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5  8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 6, 2016.

M. Thomas, Rateindependent evolution of sets & application to fracture processes, Seminar on Analysis, Kanazawa University, Institute of Science and Engineering, Kanazawa, Japan, October 28, 2016.

N. Ahmed, On the graddiv stabilization for the steady Oseen and NavierStokes evaluations, International Conference of Boundary and Interior Layers (BAIL 2016), August 15  19, 2016, Beijing Computational Science Research Center, Beijing, China, August 15, 2016.

P.É. Druet, Existence of global weak solutions for generalized PoissonNernstPlanck systems, 7th European Congress of Mathematics (ECM), minisymposium ``Analysis of Thermodynamically Consistent Models of Electrolytes in the Context of Battery Research'', July 18  22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

S.P. Frigeri, On a diffuse interface model of tumor growth, 9th European Conference on Elliptic and Parabolic Problems, May 23  27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, June 9, 2016.

M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinitedimensional Systems, March 14  18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

A. Mielke, Entropyentropy production estimates for energyreaction diffusion systems, Workshop ``Forefront of PDEs: Modelling, Analysis and Numerics'', December 12  14, 2016, Technische Universität Wien, Institut für Analysis and Scientific Computing, Austria, December 13, 2016.

A. Mielke, Evolution driven by energy and entropy, SFB1114 Kolloquium, Freie Universität Berlin, Berlin, June 30, 2016.

A. Mielke, Evolutionary Gammaconvergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29  September 2, 2016, HumboldtUniversität zu Berlin, Institut für Mathematik.

A. Mielke, Exponential decay into thermodynamical equilibrium for reactiondiffusion systems with detailed balance, Workshop ``Patterns of Dynamics'', July 25  29, 2016, Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 28, 2016.

A. Mielke, Global existence for finitestrain viscoplasticity via the energydissipation principle, Seminar ``Analysis & Mathematical Physics'', Institute of Science and Technology Austria (IST Austria), Vienna, Austria, July 7, 2016.

A. Mielke, Gradient structures and dissipation distances for reactiondiffusion equation, Mathematisches Kolloquium, Westfälische WilhelmsUniversität, Institut für Mathematik, Münster, April 28, 2016.

A. Mielke, Mutual recovery sequences and evolutionary relaxation of a twophase problem, 2nd Workshop on CENTRAL Trends in Analysis and Numerics for PDEs, May 26  28, 2016, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, May 27, 2016.

A. Mielke, On the HellingerKantorovich distance for reaction and diffusion, Workshop ``Interactions between Partial Differential Equations & Functional Inequalities'', September 12  16, 2016, The Royal Swedish Academy of Sciences, Institut MittagLeffler, Stockholm, Sweden, September 12, 2016.

A. Mielke, On the geometry of reaction and diffusion, INdAMISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5  8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 7, 2016.

A. Mielke, Optimal transport versus reaction  On the geometry of reactiondiffusion equations, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 12, 2016.

J. Rehberg, On Hölder continuity for elliptic and parabolic problems, 8th Singular Days, June 27  30, 2016, University of Lorraine, Department of Sciences and Technologies, Nancy, France, June 29, 2016.

J. Rehberg, On nonsmooth parabolic equations, Oberseminar Analysis, Universität Kassel, Institut für Mathematik, May 2, 2016.

J. Rehberg, On nonsmooth parabolic equations, Oberseminar Analysis, Leibniz Universität Hannover, Institut für Angewandte Mathematik, May 10, 2016.

J. Rehberg, On optimal elliptic Sobolev regularity, Oberseminar Analysis, Technische Universität Clausthal, Institut für Mathematik, ClausthalZellerfeld, January 20, 2016.

E. Valdinoci, Nonlocal equations in phase transitions, minimal surfaces, crystallography and life sciences, Escuela CAPDE Ecuaciones diferenciales parciales no lineales, Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Santiago de Chile, Chile.

E. Cinti, A quantitative weighted isoperimetric inequality via the ABP method, Oberseminar Analysis, Universität Bonn, Institut für Angewandte Mathematik, February 5, 2015.

E. Cinti, Quantitative flatness results for nonlocal minimal surfaces in low dimensions, Theory of Applications of Partial Differential Equations (PDE 2015), November 30  December 4, 2015, WIAS Berlin, Berlin, December 2, 2015.

E. Cinti, Quantitative isoperimetric inequality via the ABP method, Università di Bologna, Dipartimento di Matematica, Bologna, Italy, July 17, 2015.

K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Mathematical Thermodynamics of Complex Fluids, June 29  July 3, 2015, Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 30, 2015.

K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Workshop ``Young Researchers in Fluid Dynamics'', June 18  19, 2015, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt, June 18, 2015.

K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Seminar ``Dynamische Systeme'', Technische Universität München, Zentrum Mathematik, München, February 2, 2015.

K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Oberseminar ``Analysis'', Universität Kassel, Institut für Mathematik, Kassel, January 12, 2015.

K. Disser, Dynamik von Starrkörpern, die mit einer Flüssigkeit gefüllt sind, und das EiProblem, Mathematisches Kolloquium, HeinrichHeine Universität Düsseldorf, Institut für Mathematik, Düsseldorf, April 10, 2015.

S. Patrizi, Dislocations dynamics: From microscopic models to macroscopic crystal plasticity, Analysis Seminar, The University of Texas at Austin, Department of Mathematics, USA, January 21, 2015.

S. Patrizi, Dislocations dynamics: From microscopic models to macroscopic crystal plasticity, Seminar, King Abdullah University of Science and Technologie, SRI  Center for Uncertainty Quantification in Computational Science & Engineering, Jeddah, Saudi Arabia, March 25, 2015.

S. Patrizi, On a long range segregation model, Seminar, Università degli Studi di Salerno, Dipartimento di Matematica, Italy, May 19, 2015.

S. Patrizi, On a long range segregation model, Seminario di Analisi Matematica, Sapienza Università di Roma, Dipartimento di Matematica ``Guido Castelnuovo'', Italy, April 20, 2015.

E. Rocca, Optimal control of a nonlocal convective CahnHilliard equation by the velocity, Numerical Analysis Seminars, Durham University, UK, March 13, 2015.

S.P. Frigeri, On a diffuse interface model of tumor growth, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 22, 2015.

S.P. Frigeri, On a nonlocal diffuse interface model for binary incompressible fluids with different densities, Mathematical Thermodynamics of Complex Fluids, June 28  July 3, 2015, Fondazione CIME ``Roberto Conti'' (International Mathematical Summer Center), Cetraro, Italy, July 2, 2015.

S.P. Frigeri, Recent results on optimal control for CahnHilliard/NavierStokes systems with nonlocal interactions, Control Theory and Related Topics, April 13  14, 2015, Politecnico di Milano, Italy, April 13, 2015.

M. Heida, Stochastic homogenization of PrandtlReuss plasticity, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 1, 2015.

CH. Heinemann, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, 3rd Workshop of the GAMM Activity Group Analysis of Partial differential Equations, September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, October 1, 2015.

CH. Heinemann, On elastic CahnHilliard systems coupled with evolution inclusions for damage processes, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Young Researchers' Minisymposium 2, March 23  27, 2015, Lecce, Italy, March 23, 2015.

CH. Heinemann, Solvability of differential inclusions describing damage processes and applications to optimal control problems, Universität EssenDuisburg, Fakultät für Mathematik, Essen, December 3, 2015.

M. Landstorfer, Theory, structure and experimental justification of the metal/electrolyte interface, Minisymposium `` Recent Developments on Electrochemical Interface Modeling'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 11, 2015.

M. Liero, Electrothermal modeling of largearea OLEDs, sc Matheon Center Days, April 20  21, 2015, Technische Universität Berlin, Institut für Mathematik, Berlin, April 20, 2015.

M. Liero, On $p(x)$Laplace thermistor models describing electrothermal feedback in organic semiconductor devices, Theory of Applications of Partial Differential Equations (PDE 2015), November 30  December 4, 2015, WIAS Berlin, Berlin, December 3, 2015.

M. Liero, On a PDE thermistor system for largearea OLEDs, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2015), March 11  13, 2015, WIAS Berlin, Berlin, March 12, 2015.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich distance, Workshop ``Collective Dynamics in Gradient Flows and Entropy Driven Structures'', June 1  5, 2015, Gran Sasso Science Institute, L'Aquila, Italy, June 3, 2015.

D. Peschka, Droplets on liquids and their long way into equilibrium, Minisymposium ``Recent Progress in Modeling and Simulation of Multiphase Thinfilm Type Problems'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

D. Peschka, Thinfilm equations with free boundaries, Jahrestagung der Deutschen MathematikerVereinigung, Minisymposium ``Mathematics of Fluid Interfaces'', September 21  25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 23, 2015.

D. Puzyrev, Multistability and bifurcations of laser cavity solitons induced by delayed feedback, International Workshop ``Nonlinear Photonics: Theory, Materials, Applications'', Session ``Laser Dynamics'', June 29  July 2, 2015, SaintPetersburg State University, Departmetnt of General Physics, Russian Federation, July 1, 2015.

E. Valdinoci, Dislocation dynamics in crystals: Nonlocal effects, collisions and relaxation, Mostly Maximum Principle, September 16  18, 2015, Castello Aragonese, Agropoli, Italy, September 16, 2015.

E. Valdinoci, Dislocation dynamics in crystals: Nonlocal effects, collisions and relaxation, Second Workshop on Trends in Nonlinear Analysis, September 24  26, 2015, GNAMPA, Universitá degli Studi die Cagliari, Dipartimento di Matematica e Informatica, Cagliari, Italy, September 26, 2015.

E. Valdinoci, Minimal surfaces and phase transitions with nonlocal interactions, Analysis Seminar, University of Edinburgh, School of Mathematics, UK, March 23, 2015.

E. Valdinoci, Nonlocal Problems in Analysis and Geometry, 2° Corso Intensivo di Calcolo delle Variazioni, June 15  20, 2015, Dipartimento di Matematica e Informatica di Catania, Italy.

E. Valdinoci, Nonlocal minimal surfaces, Seminario di Calcolo delle Variazioni & Equazioni alle Derivate Parziali, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Italy, March 13, 2015.

E. Valdinoci, Nonlocal problems  Theory and applications, School/Workshop ``Phase Transition Problems and Nonlinear PDEs'', March 9  11, 2015, Università di Bologna, Dipartimento di Matematica.

E. Valdinoci, Nonlocal problems and applications, Summer School on ``Geometric Methods for PDEs and Dynamical Systems'', June 8  11, 2015, École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées and Institut de Mathématiques, Equipe d'Analyse, Université Bordeaux 1, Porquerolles, France.

E. Valdinoci, Some models arising in crystal dislocations, Global Dynamics in Hamiltonian Systems, June 28  July 4, 2015, Universitat Politècnica de Catalunya (BarcelonaTech), Girona, Spain, June 29, 2015.

E. Valdinoci, What is the (fractional) Laplacian?, PerlenKolloquium, Universität Basel, Fachbereich Mathematik, Switzerland, May 22, 2015.

S. Yanchuk, Multiscale jittering in oscillators with pulsatile delayed feedback, Short Thematic Program on Delay Differential Equations, May 11  15, 2015, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, May 14, 2015.

S. Yanchuk, Pattern formation in systems with multiple delayed feedbacks, Minisymposium ``Timedelayed Feedback'' of the SIAM Conference on Applications of Dynamical Systems, May 17  21, 2015, Society for Industrial and Applied Mathematics, Snowbird/Utah, USA, May 20, 2015.

S. Yanchuk, Stability of plane wave solutions in complex GinzburgLandau equation with delayed feedback, Jahrestagung der Deutschen MathematikerVereinigung, Minisymposium ``Topics in Delay Differential Equations'', September 21  25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 22, 2015.

S. Yanchuk, Time delays and plasticity in neuronal networks, Seminar ``Selfassembly and Selforganization in Computer Science and Biology'', September 27  October 2, 2015, LeibnizZentrum für Informatik, Schloss Dagstuhl, September 28, 2015.

J. Sprekels, Optimal boundary control problems for CahnHilliard systems with dynamic boundary conditions, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 21, 2015.

A. Glitzky, Analysis of $p(x)$Laplace thermistor models for electrothermal feedback in organic semiconductor devices, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, September 30, 2015.

A. Glitzky, Finite volume discretized reactiondiffusion systems in heterostructures, Conference on Partial Differential Equations, March 25  29, 2015, Technische Universität München, Zentrum Mathematik, München, March 28, 2015.

M. Thomas, Analysis for edgeemitting semiconductor heterostructures, Minisymposium ``Numerical and Analytical Aspects in Semiconductor Theory'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 2015.

M. Thomas, Analysis of nonsmooth PDE systems with application to material failuretowards dynamic fracture, Minisymposium ``Analysis of Nonsmooth PDE Systems with Application to Material Failure'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10  14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Pavia, Italy, March 5, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Evolutionary Gammaconvergence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Session on Applied Analysis, March 23  27, 2015, Università del Salento, Lecce, Italy, March 26, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence and evolutionary Gamma convergence, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 19, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), GAMM Juniors Poster Session, Lecce, Italy, March 23  27, 2015.

M. Thomas, Evolutionary Gamma convergence with application to damage and delamination, Seminar DICATAM, Università di Brescia, Dipartimento di Matematica, Brescia, Italy, June 3, 2015.

M. Thomas, Rateindependent damage models with spatial BVregularization  Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

M.H. Farshbaf Shaker, A deep quench approach to the optimal control of an AllenCahn equation with dynamic boundary conditions, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 20, 2015.

M.H. Farshbaf Shaker, Multimaterial phase field approach to structural topology optimization and its relation to sharp interface approach, University of Tokyo, Graduate School of Mathematical Sciences, Japan, October 6, 2015.

M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 13, 2015.

M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, Technische Universität Berlin, Institut für Mathematik, February 5, 2015.

CH. Heinemann, Wellposedness of strong solutions for a damage model in 2D, Universitá di Brescia, Department DICATAM  Section of Mathematics, Italy, March 13, 2015.

A. Mielke, A mathematical approach to finitestrain viscoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 20, 2015.

A. Mielke, Abstract approach to energetic solutions for rateindependent solutions, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 18, 2015.

A. Mielke, Evolutionary $Gamma$convergence for generalized gradient systems, Workshop ``Gradient Flows'', June 22  23, 2015, Université Pierre et Marie Curie, Laboratoire JacquesLouis Lions, Paris, France, June 22, 2015.

A. Mielke, Evolutionary relaxation of a twophase model, MiniWorkshop ``Scales in Plasticity'', November 8  14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

A. Mielke, Existence results in finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 19, 2015.

A. Mielke, Geometric approaches at and for theoretical and applied mechanics, Phil Holmes Retirement Celebration, October 8  9, 2015, Princeton University, Mechanical and Aerospace Engineering, New York, USA, October 8, 2015.

A. Mielke, Mathematical modeling for finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 16, 2015.

A. Mielke, The Chemical Master Equation as entropic gradient flow, Conference ``New Trends in Optimal Transport'', March 2  6, 2015, Universität Bonn, Institut für Angewandte Mathematik, March 2, 2015.

A. Mielke, The FokkerPlanck and Liouville equations for chemical reactions as largevolume approximations of the Chemical Master Equation, Workshop ``Stochastic Limit Analysis for Reacting Particle Systems'', December 16  18, 2015, WIAS Berlin, Berlin, December 18, 2015.

A. Mielke, The multiplicative strain decomposition in finitestrain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16  20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 17, 2015.

H. Neidhardt, Trace formulas for nonadditive perturbations, Conference ``Spectral Theory and Applications'', May 25  29, 2015, AGH University of Science and Technology, Krakow, Poland, May 28, 2015.

J. Rehberg, On maximal parabolic regularity, The Fourth Najman Conference on Spectral Problems for Operators and Matrices, September 20  25, 2015, University of Zagreb, Department of Mathematics, Opatija, Croatia, September 23, 2015.

J. Rehberg, On maximal parabolic regularity and its applications, Oberseminar ``Mathematische Optimierung'', Technische Universität München, Lehrstuhl für Optimalsteuerung, München, May 6, 2015.

K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Second Workshop of the GAMM Activity Group on "Analysis of Partial Differential Equations", September 29  October 1, 2014, Universität Stuttgart, Lehrstuhl für Analysis und Modellierung, October 1, 2014.

K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Autumn School and Workshop on Mathematical Fluid Dynamics, October 27  30, 2014, Universität Darmstadt, International Research Training Group 1529, Bad Boll, October 28, 2014.

K. Disser, Parabolic equations with mixed boundary conditions, degenerate diffusion and diffusion on Lipschitz interfaces, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Operator Theory, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 14, 2014.

K. Disser, Parabolic equations with mixed boundary conditions, degenerate diffusion and diffusion on Lipschitz interfaces, International Conference ``Vorticity, Rotations and Symmetry (III)Approaching Limiting Cases of Fluid Flow'', May 5  9, 2014, Centre International de Rencontres Mathématiques (CIRM), Luminy, Marseille, France.

C. Kreisbeck, Thinfilm limits of functionals on Afree vector fields and applications, Workshop on Trends in NonLinear Analysis 2014, July 31  August 1, 2014, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, August 1, 2014.

C. Kreisbeck, Thinfilm limits of functionals on Afree vector fields and applications, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8  11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

C. Kreisbeck, Thinfilm limits of functionals on Afree vector fields and applications, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, July 16, 2014.

S. Reichelt, Effective equations for reactiondiffusion systems in strongly heterogeneous media, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 10, 2014.

S. Reichelt, Twoscale homogenization of nonlinear reactiondiffusion systems involving different diffusion length scales, scshape Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, December 3, 2014.

S. Reichelt, Twoscale homogenization of reactiondiffusion systems with small diffusion, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

S. Reichelt, Twoscale homogenization of reactiondiffusion systems with small diffusion, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7  11, 2014, Madrid, Spain, July 8, 2014.

E. Rocca, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, Symposium on Trends in Application of Mathematics to Mechanics (STAMM), September 8  11, 2014, International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

S. Heinz, Analysis and numerics of a phasetransformation model, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich distance, Workshop ``Entropy Methods, PDEs, Functional Inequalities, and Applications'', June 30  July 4, 2014, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, July 1, 2014.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich distance, RIPE60  Rate Independent Processes and Evolution Workshop, June 24  26, 2014, Prague, Czech Republic, June 24, 2014.

S. Neukamm, Optimal quantitative twoscale expansion in stochastic homogenization, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

D.R.M. Renger, Connecting particle systems to entropydriven gradient flows, Conference on Nonlinearity, Transport, Physics, and Patterns, October 6  10, 2014, Fields Institute for Research in Mathematical Sciences, Toronto, Canada, October 9, 2014.

D.R.M. Renger, Connecting particle systems to entropydriven gradient flows, Oberseminar ``Stochastische und Geometrische Analysis'', Universität Bonn, Institut für Angewandte Mathematik, May 28, 2014.

E. Valdinoci, (Non)local interfaces and minimal surfaces, International Conference on ``Nonlinear Phenomena in Biology'', March 5  7, 2014, Helmholtz Zentrum München  Deutsches Forschungszentrum für Gesundheit und Umwelt, March 5, 2014.

E. Valdinoci, Concentrating solutions for a nonlocal Schroedinger equation, Nonlinear Partial Differential Equations and Stochastic Methods, June 7  11, 2014, University of Jyväskylä, Finland, June 10, 2014.

E. Valdinoci, Concentration phenomena for nonlocal equation, Méthodes Géométriques et Variationnelles pour des EDPs Nonlinéaires, September 1  5, 2014, Université C. Bernard, Lyon 1, Institut C. Jordan, France, September 2, 2014.

E. Valdinoci, Concentration solutions for a nonlocal Schroedinger equation, Kinetics, Non Standard Diffusion and the Mathematics of Networks: Emerging Challenges in the Sciences, May 7  16, 2014, The University of Texas at Austin, Department of Mathematics, USA, May 14, 2014.

E. Valdinoci, Dislocation dynamics and fractional equations, Analysis Seminar, HeriotWatt University of Edinburgh, London, UK, October 31, 2014.

E. Valdinoci, Dislocation dynamics and fractional equations, Analysis Seminar, University of Texas at Austin Mathematics, USA, November 5, 2014.

E. Valdinoci, Dislocation dynamics in crystals, Recent Advances in Nonlocal and Nonlinear Analysis: Theory and Applications, June 10  14, 2014, FIM  Institute for Mathematical Research, ETH Zuerich, Switzerland, June 13, 2014.

E. Valdinoci, Dislocation dynamics in crystals, Geometry and Analysis Seminar, Columbia University, Department of Mathematics, New York City, USA, April 3, 2014.

E. Valdinoci, Dislocation dynamics in crystals, Seminari di Analisi Matematica, Università di Torino, Dipartimento di Matematica ``Giuseppe Peano'', Italy, December 18, 2014.

E. Valdinoci, Gradient estimates and symmetry results in anisotropic media, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 76: Viscosity, Nonlinearity and Maximum Principle, July 7  11, 2014, Madrid, Spain, July 8, 2014.

E. Valdinoci, Nonlinear PDEs, Spring School on Nonlinear PDEs, March 24  27, 2014, INdAM Istituto Nazionale d'Alta Matematica, Sapienza  Università di Roma, Italy.

E. Valdinoci, Nonlocal equations and applications, Seminario de Ecuaciones Diferenciales, Universidad de Granada, IEMathGranada, Spain, November 28, 2014.

E. Valdinoci, Nonlocal minimal surfaces, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 96: Geometric Variational Problems with Associated Stability Estimates, July 7  11, 2014, Madrid, Spain, July 8, 2014.

E. Valdinoci, Nonlocal minimal surfaces and free boundary problems, Geometric Aspects of Semilinear Elliptic and Parabolic Equations: Recent Advances and Future Perspectives, May 25  30, 2014, Banff International Research Station for Mathematical Innovation and Discovery, Calgary, Canada, May 27, 2014.

E. Valdinoci, Nonlocal problems in analysis and geometry, December 1  5, 2014, Universidad Autonoma de Madrid, Departamento de Matemáticas, Spain.

E. Valdinoci, Some nonlocal aspects of partial differential equations and free boundary problems, Institutskolloquium, Weierstrass Institut Berlin (WIAS), January 13, 2014.

A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reactiondiffusion systems, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin, June 15  20, 2014.

A. Glitzky, Driftdiffusion models for heterostructures in photovoltaics, 8th European Conference on Elliptic and Parabolic Problems, Minisymposium ``Qualitative Properties of Nonlinear Elliptic and Parabolic Equations'', May 26  30, 2014, Universität Zürich, Institut für Mathematik, organized in Gaeta, Italy, May 27, 2014.

D. Knees, A quasilinear differential inclusion for viscous and rateindependent damage systems in nonsmooth domains, Analysis & Stochastics Seminar, Technische Universität Dresden, Institut für Analysis, January 16, 2014.

M. Thomas, A stressdriven localsolution approach to quasistatic brittle delamination, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 91: Variational Methods for Evolution Equations, July 7  11, 2014, Madrid, Spain, July 7, 2014.

M. Thomas, Existence & stability results for rateindependent processes in viscoelastic materials, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, March 18, 2014.

M. Thomas, Existence and stability results for rateindependent processes in viscoelastic materials, Women in Partial Differential Equations & Calculus of Variations Workshop, March 6  8, 2014, University of Oxford, Mathematical Institute, UK, March 6, 2014.

M. Thomas, GENERIC for solids with dissipative interface processes, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), GAMM Juniors' Poster Session, FriedrichAlexander Universität ErlangenNürnberg, March 10  14, 2014.

M. Thomas, Rateindependent systems with viscosity and inertia: Existence and evolutionary Gammaconvergence, Workshop ``Variational Methods for Evolution'', December 14  20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, International Workshop ``Variational Modeling in Solid Mechanics'', September 22  24, 2014, University of Udine, Department of Mathematics and Informatics, Italy, September 23, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

M. Thomas, Stressdriven localsolution approach to quasistatic brittle delamination, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 11, 2014.

B. Wagner, Asymptotic analysis of interfacial evolution, BMSWIAS Summer School ``Applied Analysis for Materials'', August 25  September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

A. Mielke, S. Reichelt, M. Thomas, Pattern formation in systems with multiple scales, Evaluation of the DFG Collaborative Research Center 910 ``Control of Selforganizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Berlin, June 30  July 1, 2014.

A. Mielke, Evolutionary Gamma convergence and amplitude equations, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 13, 2014.

A. Mielke, Generalized gradient structures for reactiondiffusion systems, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, June 17, 2014.

A. Mielke, Gradient structures and dissipation distances for reactiondiffusion systems, Workshop ``Advances in Nonlinear PDEs: Analysis, Numerics, Stochastics, Applications'', June 2  3, 2014, Vienna University of Technology and University of Vienna, Austria, June 2, 2014.

A. Mielke, Gradient structures and dissipation distances for reactiondiffusion systems, Seminar ``Analysis of Fluids and Related Topics'', Princeton University, Department of Mechanical and Aerospace Engineering, Princeton, NJ, USA, March 6, 2014.

A. Mielke, Homogenization of parabolic gradient systems via evolutionary $Gamma$convergence, Second Workshop of the GAMM Activity Group on ``Analysis of Partial Differential Equations'', September 29  October 1, 2014, Universität Stuttgart, Institut für Analysis, Dynamik und Modellierung, September 30, 2014.

A. Mielke, How thermodynamics induces geometry structures for reactiondiffusion systems, Gemeinsames Mathematisches Kolloquium der Universitäten Gießen und Marburg, Universität Gießen, Mathematisches Institut, January 15, 2014.

A. Mielke, Multiscale modeling and evolutionary Gammaconvergence for gradient flows, BMSWIAS Summer School ``Applied Analysis for Materials'', August 25  September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

A. Mielke, On a metric and geometric approach to reactiondiffusion systems as gradient systems, Mathematics Colloquium, Jacobs University Bremen, School of Engineering and Science, December 1, 2014.

A. Mielke, On gradient structures and dissipation distances for reactiondiffusion systems, Kolloquium ``Angewandte Mathematik'', FriedrichAlexanderUniversität ErlangenNürnberg, Department Mathematik, July 3, 2014.

A. Mielke, On gradient structures for reactiondiffusion systems, Joint Analysis Seminar, RheinischWestfälische Technische Hochschule Aachen (RWTH), Institut für Mathematik, February 4, 2014.

A. Mielke, On the microscopic origin of generalized gradient structures for reactiondiffusion systems, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8  11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 11, 2014.

A. Mielke, A reactiondiffusion equation as a HellingerKantorovich gradient flow, ERC Workshop on Optimal Transportation and Applications, October 27  31, 2014, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, October 29, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties in elasticity with homogenization, 4th BritishGerman Frontiers of Science Symposium, Potsdam, March 6  9, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties with homogenization, 4th BritishGerman Frontiers of Science Symposium, March 6  9, 2014, Alexander von HumboldtStiftung, Potsdam, March 7, 2014.

S. Neukamm, Homogenization of nonlinear bending plates, Workshop ``Relaxation, Homogenization, and Dimensional Reduction in Hyperelasticity'', March 25  27, 2014, Université ParisNord, France, March 26, 2014.

S. Neukamm, Homogenization of slender structures in smallstrain regimes, 14th Dresden Polymer Discussion, Meißen, May 25  28, 2014.

J. Rehberg, Maximal parabolic regularity on strange geometries and applications, Joint Meeting 2014 of the German Mathematical Society (DMV) and the Polish Mathematical Society (PTM), September 17  20, 2014, Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Poznan, Poland, September 18, 2014.

J. Rehberg, On nonsmooth parabolic equations, Workshop ``MaxwellStefan meets NavierStokes/Modeling and Analysis of Reactive MultiComponent Flows'', March 31  April 2, 2014, Universität Halle, April 1, 2014.

J. Rehberg, Optimal Sobolev regularity for second order divergence operators, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Operator Theory, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 13, 2014.

K. Disser, Entropic gradient structures for reversible Markov chains and the passage to Wasserstein FokkerPlanck, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1  2, 2013.

K. Disser, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

K. Disser, Passage to the limit of the entropic gradient structure of reversible Markov processes to the Wasserstein FokkerPlanck equation, Oberseminar Analysis, MartinLutherUniversität HalleWittenberg, Institut für Mathematik, Halle, November 20, 2013.

K. Götze, Free fall of a rigid body in a viscoelastic fluid, Workshop ``Geophysical Fluid Dynamics'', February 18  22, 2013, Mathematisches Forschungsinstitut Oberwolfach, February 20, 2013.

K. Götze, Maximal regularity for fluidrigid body interaction problems, Interim Evaluation of the International Research Training Network 1529 ``Mathematical Fluid Dynamics'', Technische Universität Darmstadt, Fachbereich Mathematik, January 11, 2013.

K. Götze, Starke Lösungen für die Bewegung von Starrkörpern in Flüssigkeiten, Oberseminar Analysis, Technische Universität Dresden, Institut für Analysis, January 24, 2013.

S. Reichelt, Homogenization of degenerated reactiondiffusion equations, Doktorandenforum der LeibnizGemeinschaft, Sektion D, Berlin, June 6  7, 2013.

S. Reichelt, Introduction to homogenization concepts, Freie Universität Berlin, Institut für Mathematik, April 11, 2013.

S. Reichelt, Twoscale homogenization of nonlinear reactiondiffusion systems with small diffusion, Workshop on Control of SelfOrganizing Nonlinear Systems, August 28  30, 2013, SFB 910 ``Control of selforganizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, August 30, 2013.

S. Reichelt, Twoscale homogenization in nonlinear reactiondiffusion systems with small diffusion, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

P. Gussmann, Linearized elasticity as $Gamma$limit of finite elasticity in the case of cracks, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Section ``Applied Analysis'', March 18  22, 2013, University of Novi Sad, Serbia, March 20, 2013.

CH. Heinemann, Analysis of degenerating CahnHilliard systems coupled with complete damage processes, 2013 CNA Summer School, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, USA, May 30  June 7, 2013.

CH. Heinemann, Degenerating CahnHilliard systems coupled with complete damage processes, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 2013.

CH. Heinemann, Degenerating CahnHilliard systems coupled with mechanical effects and complete damage processes, Equadiff13, MS27  Recent Results in Continuum and Fracture Mechanics, August 26  30, 2013, Prague, Czech Republic, August 27, 2013.

CH. Heinemann, On a PDE system describing damage processes and phase separation, Oberseminar Analysis, Universität Augsburg, July 11, 2013.

S. Heinz, On a way to control oscillations for a special evolution equation, Conference ``Nonlinearities'', June 10  14, 2013, University of Warsaw, Institute of Mathematics, Male Ciche, Poland, June 11, 2013.

M. Liero, On gradient structures for driftreactiondiffusion systems and Markov chains, Analysis Seminar, University of Bath, Mathematical Sciences, UK, November 21, 2013.

M. Liero, Gradient structures and geodesic convexity for reactiondiffusion system, SIAM Conference on Mathematical Aspects of Materials Science (MS13), Minisymposium ``Material Modelling and Gradient Flows'' (MS100), June 9  12, 2013, Philadelphia, USA, June 12, 2013.

M. Liero, On gradient structures and geodesic convexity for reactiondiffusion systems, Research Seminar, Westfälische WilhelmsUniversität Münster, Institut für Numerische und Angewandte Mathematik, April 17, 2013.

S. Neukamm, Quantitative results in stochastic homogenization, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, June 27, 2013.

S. Neukamm, Quantitative results in stochastic homogenization, Oberseminar Analysis, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, June 13, 2013.

S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, GermanJapanese Meeting on Stochastic Analysis, September 9  13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the NavierStokesKorteweg model, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 2013.

A. Glitzky, Continuous and finite volume discretized reactiondiffusion systems in heterostructures, Asymptotic Behaviour of Systems of PDE Arising in Physics and Biology: Theoretical and Numerical Points of View, November 6  8, 2013, Lille 1 University  Science and Technology, France, November 6, 2013.

A. Glitzky, Nonlinear electrothermal feedback in organic semiconductors, Organic Photovoltaics Workshop 2013, December 10  11, 2013, University of Oxford, Mathematical Insitute, UK, December 10, 2013.

D. Knees, A vanishing viscosity approach to a rateindependent damage model, Seminar ``Wissenschaftliches Rechnen'', Technische Universität Dortmund, Fachbereich Mathematik, January 31, 2013.

D. Knees, Crack evolution models based on the Griffith criterion, Workshop on Mathematical Aspects of Continuum Mechanics, October 12  14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 13, 2013.

D. Knees, Global spatial regularity for elasticity models with cracks and contact, Journées Singulières Augmentées 2013, August 26  30, 2013, Université de Rennes 1, France, August 27, 2013.

D. Knees, Global spatial regularity results for crack with contact and application to a fracture evolution model, Oberseminar Nichtlineare Analysis, Universität Köln, Mathematisches Institut, October 28, 2013.

D. Knees, Modeling and analysis of crack evolution based on the Griffith criterion, Nonlinear Analysis Seminar, Keio University of Science, Yokohama, Japan, October 9, 2013.

D. Knees, On energy release rates for nonlinearly elastic materials, Workshop on Mathematical Aspects of Continuum Mechanics, October 12  14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 12, 2013.

D. Knees, Recent results in nonlinear elasticity and fracture mechanics, Universität der Bundeswehr, Institut für Mathematik und Bauinformatik, München, August 13, 2013.

D. Knees, Weak solutions for rateindependent systems illustrated at an example for crack propagation, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013  Diffuse Interface Models, September 10  13, 2013, Levico Terme, Italy, September 11, 2013.

CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

CH. Kraus, Sharp interface limit of a diffuse interface model of NavierStokesAllenCahn type for mixtures, Workshop ``Hyperbolic Techniques for Phase Dynamics'', June 10  14, 2013, Mathematisches Forschungsinstitut Oberwolfach, June 11, 2013.

M. Thomas, A stressdriven local solution approach to quasistatic brittle delamination, BMS Intensive Course on Evolution Equations and their Applications, November 27  29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 29, 2013.

M. Thomas, A stressdriven local solution approach to quasistatic brittle delamination, Seminar on Functional Analysis and Applications, International School of Advanced Studies (SISSA), Trieste, Italy, November 12, 2013.

M. Thomas, Existence & fine properties of solutions for rateindependent brittle damage models, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1  2, 2013.

M. Thomas, Damage and delamination processes in thermoviscoelastic materials, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Young Reserchers' Minisymposium ``Analytical and Engineering Aspects in the Material Modeling of Solids'', March 18  22, 2013, University of Novi Sad, Serbia, March 19, 2013.

M. Thomas, Existence & fine properties of solutions for rateindependent brittle damage models, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM Juniors Poster Exhibition, Novi Sad, Serbia, March 18  22, 2013.

M. Thomas, Fine properties of solutions for rateindependent brittle damage models, XXIII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, Italy, February 3  8, 2013.

M. Thomas, Local versus energetic solutions in rateindependent brittle delamination, DIMO2013  Diffuse Interface Models, September 10  13, 2013, Levico Terme, Italy, September 13, 2013.

M. Thomas, Rateindependent damage models with spatial BVregularization  Existence & fine properties of solutions, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 24, 2013.

H. Hanke, Derivation of an effective damage model with evolving microstructure, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 29, 2013.

H. Hanke, Derivation of an effective damage model with nonperiodic evolving microstructure, 12th GAMM Seminar on Microstructures, February 8  9, 2013, HumboldtUniversität zu Berlin, Institut für Mathematik, February 9, 2013.

A. Mielke, Gradient structures and dissipation distances for reactiondiffusion systems, Workshop ``Material Theory'', December 16  20, 2013, Mathematisches Forschungsinstitut Oberwolfach, December 17, 2013.

A. Mielke, Analysis, modeling, and simulation of semiconductor devices, Kolloquium Simulation Technology, Universität Stuttgart, SRC Simulation Technology, May 14, 2013.

A. Mielke, Deriving the GinzburgLandau equation as amplitude equation via evolutionary Gamma convergence, ERC Workshop on Variational Views on Mechanics and Materials, June 24  26, 2013, University of Pavia, Department of Mathematics, Italy, June 26, 2013.

A. Mielke, Evolutionary Gamma convergence and amplitude equations, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, April 8, 2013.

A. Mielke, Gradient structures and uniform global decay for reactiondiffusion systems, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, April 25, 2013.

A. Mielke, On entropydriven dissipative quantum mechanical systems, Analysis and Stochastics in Complex Physical Systems, March 20  22, 2013, Universität Leipzig, Mathematisches Institut, March 21, 2013.

A. Mielke, On the geometry of reactiondiffusion systems: Optimal transport versus reaction, Recent Trends in Differential Equations: Analysis and Discretisation Methods, November 7  9, 2013, Technische Universität Berlin, Institut für Mathematik, November 9, 2013.

A. Mielke, Rateindependent plasticity as vanishingviscosity limit for wiggly energy landscape, Workshop on Evolution Problems for Material Defects: Dislocations, Plasticity, and Fracture, September 30  October 4, 2013, International School of Advanced Studies (SISSA), Trieste, Italy, September 30, 2013.

A. Mielke, Using gradient structures for modeling semiconductors, Eindhoven University of Technology, Institute for Complex Molecular Systems, Netherlands, February 21, 2013.

J. Rehberg, Sobolev extention and analysis on nonLipschitz domain, Oberseminar Angewandte Analysis, Universität Darmstadt, Fachbereich Mathematik, May 14, 2013.

J. Sprekels, Optimal control of AllenCahn equations with singular potentials and dynamic boundary conditions, DIMO2013  Diffuse Interface Models, September 10  13, 2013, Levico Terme, Italy, September 11, 2013.

J. Sprekels, Optimal control of the AllenCahn equation with dynamic boundary condition and double obstacle potentials: A ``deep quench'' approach, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 17, 2013.

J. Sprekels, PrandtlIshlinskii operators and elastoplasticity, Spring School on ``Rateindependent Evolutions and Hysteresis Modelling'', May 27  31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

K. Götze, Free fall of a rigid body in a viscoelastic fluid, International Summer School on Evolution Equations (EVEQ 2012), Prague, Czech Republic, July 9  13, 2012.

K. Götze, Free fall of a rigid body in a viscoelastic fluid, International Summer School on Evolution Equations (EVEQ 2012), July 9  13, 2012, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

K. Götze, Some results for the motion of a rigid body with a liquidfilled cavity, Seminar on Partial Differential Equations, Czech Academy of Sciences, Mathematical Institute, Prague, Czech Republic, November 6, 2012.

CH. Heinemann, Complete damage in linear elastic materials, Variational Models and Methods for Evolution, Levico, Italy, September 10  12, 2012.

CH. Heinemann, Damage processes coupled with phase separation in elastically stressed alloys, GAMM Jahrestagung 2012 (83rd Annual Meeting), March 26  30, 2012, Technische Universität Darmstadt, March 27, 2012.

CH. Heinemann, Existence of weak solutions for ratedependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, LeibnizDoktorandenForum der Sektion D, Berlin, June 7  8, 2012.

S. Heinz, Quasiconvexity equals rankone convexity for isotropic sets of 2x2 matrices, 11th GAMM Seminar on Microstructures, January 20  21, 2012, Universität DuisburgEssen, January 20, 2012.

S. Heinz, Regularization and relaxation of timecontinuous problems in plasticity, 11th GAMM Seminar on Microstructures, Universität DuisburgEssen, January 20  21, 2012.

S. Heinz, Rigorous derivation of a dissipation for laminate microstructures, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Material Modelling in Solid Mechanics, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

M. Liero, Interfaces in reactiondiffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 2012.

M. Liero, Variational methods for evolution, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

A. Glitzky, An electronic model for solar cells taking into account active interfaces, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24  28, 2012, WIAS Berlin, September 27, 2012.

A. Glitzky, Mathematische Modellierung und Simulation organischer Halbleiterbauelemente, Senatsausschuss Wettbewerb (SAW), Sektion D der LeibnizGemeinschaft, LeibnizInstitut für Analytische Wissenschaften (ISAS), Dortmund, September 14, 2012.

D. Knees, A vanishing viscosity approach in fracture mechanics, Nonlocal Models and Peridynamics, November 5  7, 2012, Technische Universität Berlin, Institut für Mathematik, November 5, 2012.

D. Knees, Modeling and mathematical analysis of elastoplastic phenomena, Winter School on Modeling Complex Physical Systems with Nonlinear (S)PDE (DoM$^2$oS), February 27  March 2, 2012, Technische Universität Dortmund, Fakultät für Mathematik.

D. Knees, On a vanishing viscosity approach in damage mechanics, Kolloquium der AG Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, Institut für Mathematik, August 21, 2012.

CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

CH. Kraus, CahnLarché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11  15, 2012, Frauenchiemsee, June 12, 2012.

CH. Kraus, Phasenfeldsysteme für Entmischungs und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

CH. Kraus, The Stefan problem with inhomogeneous and anisotropic GibbsThomson law, 6th European Congress of Mathematics, July 2  6, 2012, Cracow, Poland, July 5, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19  21, 2012, University of Udine, Department of Mathematics, Italy, September 21, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17  21, 2012, Cortona, Italy, September 17, 2012.

M. Thomas, Analytical aspects of rateindependent damage models with spatial BVregularization, Seminar, SISSA  International School for Advanced Studies, Functional Analysis and Applications, Trieste, Italy, November 28, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11  15, 2012, Universität Regensburg, Frauenchiemsee, June 11, 2012.

M. Thomas, Delamination in viscoelastic materials with thermal effects, Seminar on Applied Mathematics, Università di Brescia, Dipartimento di Matematica, Italy, March 14, 2012.

M. Thomas, Mathematical methods in continuum mechanics of solids, COMMAS (Computational Mechanics of Materials and Structures) Summer School, October 8  12, 2012, Universität Stuttgart, Institut für Mechanik (Bauwesen).

M. Thomas, Modellierung und Analysis von Delaminationsprozessen, Sitzung des Wissenschaftlichen Beirats des WIAS, Berlin, October 5, 2012.

M. Thomas, Rateindependent evolution of sets, Variational Models and Methods for Evolution, Levico, Italy, September 10  12, 2012.

M. Thomas, Thermomechanical modeling via energy and entropy using GENERIC, Workshop ``Mechanics of Materials'', March 19  23, 2012, Mathematisches Forschungsinstitut Oberwolfach, March 22, 2012.

A. Fiebach, A. Glitzky, A. Linke, Voronoi finitevolume methods for reactiondiffusionsystems, 33. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo 2012), Universität Rostock, Institut für Mathematik, May 4  5, 2012.

H. Hanke, Derivation of an effective damage evolution model, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

H. Hanke, Derivation of an effective damage evolution model using twoscale convergence techniques, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Damage Processes and Contact Problems, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 29, 2012.

H. Hanke, Derivation of an effective damage evolution model using twoscale convergence techniques, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19  21, 2012, University of Udine, Department of Mathematics, Italy, September 19, 2012.

A. Mielke, Entropy gradient flows for Markow chains and reactiondiffusion systems, BerlinLeipzigSeminar ``Analysis/Probability Theory'', WIAS Berlin, April 13, 2012.

A. Mielke, Finitestrain viscoelasticity as a gradient flow, Analysis and Applications of PDEs: An 80th Birthday Meeting for Robin Knops, December 10  11, 2012, International Center for Mathematical Sciences, Edinburgh, UK, December 11, 2012.

A. Mielke, Gamma convergence and evolution, International Conference ``Trends in Mathematical Analysis'', March 1  3, 2012, Politecnico di Milano, Dipartimento di Matematica ``F. Brioschi'', Italy, March 1, 2012.

A. Mielke, GradientenStrukturen und geodätische Konvexität für MarkovKetten und ReaktionsDiffusionsSysteme, Augsburger Kolloquium, Universität Augsburg, Institut für Mathematik, May 8, 2012.

A. Mielke, Linearized elastoplasticity is the evolutionary Gamma limit of finite elastoplasticity, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26  30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

A. Mielke, Multidimensional modeling and simulation of optoelectronic devices, Challenge Workshop ``Modeling, Simulation and Optimisation Tools'', September 24  26, 2012, Technische Universität Berlin, September 24, 2012.

A. Mielke, Multiscale modeling for evolutionary systems via Gamma convergence, NDNS$^+$ Summer School in Applied Analysis, June 18  20, 2012, University of Twente, Applied Analysis & Mathematical Physics, Enschede, Netherlands.

A. Mielke, On consistent couplings of quantum mechanical and dissipative systems, Jahrestagung der Deutschen MathematikerVereinigung (DMV) 2012, Minisymposium ``Dynamical Systems'', September 17  20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 19, 2012.

A. Mielke, On geodesic convexity for reactiondiffusion systems, Seminar on Applied Mathematics, Università di Pavia, Dipartimento di Matematica, Italy, March 6, 2012.

A. Mielke, On gradient structures and geodesic convexity for energyreactiondiffusion systems and Markov chains, ERC Workshop on Optimal Transportation and Applications, November 5  9, 2012, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, November 8, 2012.

A. Mielke, On gradient structures for Markov chains and reactiondiffusion systems, Applied & Computational Analysis (ACA) Seminar, University of Cambridge, Department of Applied Mathematics and Theoretical Physics (DAMTP), UK, June 14, 2012.

A. Mielke, Using gradient structures for modeling semiconductors, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24  28, 2012, WIAS Berlin, September 24, 2012.

J. Rehberg, Hölderstetigkeit für Lösungen elliptischer Gleichungen, Seminar der Arbeitsgruppe ``Analysis'', MartinLutherUniversität HalleWittenberg, Institut für Mathematik, December 12, 2012.

J.A. Griepentrog, On nonlocal phase separation processes in multicomponent systems, 10th GAMM Seminar on Microstructures, January 20  22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

S. Heinz, A model for the texture evolution in polycrystalline materials, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Solid Mechanics, April 18  21, 2011, Technische Universität Graz, Austria, April 19, 2011.

A. Petrov, Vibrations with unilateral constraints: An overview of M. Schatzman's contributions  Part II: Deformable bodies, 7th International Congress on Industrial and Applied Mathematics, Minisymposium ``Vibrations with Unilateral Constraints'', July 18  22, 2011, Society for Industrial and Applied Mathematics, Vancouver, Canada, July 22, 2011.

J.A. Griepentrog, The role of nonsmooth regularity theory in the analysis of phase separation processes, Ehrenkolloquium anlässlich des 60. Geburtstages von PD Dr. habil. Lutz Recke, HumboldtUniversität zu Berlin, Institut für Mathematik, November 21, 2011.

S. Heinz, Regularizations and relaxations of timecontinuous problems in plasticity, Workshop der Forschergruppe 797 ``Analysis and Computation of Microstructure in Finite Plasticity'', Universität Bonn, Mathematisches Institut, November 14, 2011.

U. Stefanelli, Evolution = Minimization?, Friday Colloquium, Berlin Mathematical School, May 27, 2011.

K. Götze, Starke Lösungen für die Interaktion von starren Körpern und viskoelastischen Flüssigkeiten, Lectures in Continuum Mechanics, Universität Kassel, Institut für Mathematik, November 7, 2011.

A. Glitzky, An electronic model for solar cells including active interfaces, Workshop ``Mathematical Modelling of Organic Photovoltaic Devices'', University of Cambridge, Department of Applied Mathematics and Theoretical Physics, UK, June 9, 2011.

CH. Heinemann, Existence results for CahnHilliard equations coupled with elasticity and damage, Workshop on Phase Separation, Damage and Fracture, September 21  23, 2011, WIAS, September 23, 2011.

D. Knees, A vanishing viscosity approach in damage mechanics, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), June 20  24, 2011, Gargnano (Brescia), Italy, June 22, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, Workshop ``Perspectives in Continuum Mechanics'' in Honor of Gianfranco Capriz's 85th Birthday, University of Florence, Department of Mathematics, Italy, January 28, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 7th International Congress on Industrial and Applied Mathematics, Session ``Materials Science II'', July 18  22, 2011, Society for Industrial and Applied Mathematics, Vancouver, Canada, July 19, 2011.

D. Knees, On a vanishing viscosity approach for a model in damage mechanics, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Applied Analysis, April 18  21, 2011, Technische Universität Graz, Austria, April 20, 2011.

CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18  22, 2011, Vancouver, Canada, July 18, 2011.

A. Mielke, Complex hysteresis operators arising from homogenization and dimension reduction, 8th International Symposium on Hysteresis Modelling and Micromagnetics (HMM2011), Session ``Mathematics of Hysteresis II'', May 9  11, 2011, Università degli Studi di Trento, Centro Internazionale per la Ricerca Matematica, Levico Terme, Italy, May 10, 2011.

A. Mielke, Multiscale problems in systems driven by functionals, ISAMTopMath Summer School 2011 on Variational Methods, September 12  16, 2011, Technische Universität München, Fakultät für Mathematik.

A. Mielke, Remarks on evolutionary multiscale systems driven by functionals, Intellectual Challenges in Multiscale Modeling of Solids, July 4  5, 2011, Oxford University, Mathematical Institute, UK, July 4, 2011.

J. Sprekels, A nonstandard phase field system of CahnHilliard type for diffusive phase segregation, Seminario Matematico e Fisico di Milano, Università degli Studi di Milano, Dipartimento di Matematica, Italy, September 21, 2011.

M. Thomas, Modeling and analysis of rateindependent damage and delamination processes, 19th International Conference on Computer Methods in Mechanics, Minisymposium ``Growth Phenomena and Evolution of Microstructures. Applications in Solids'', May 9  12, 2011, Warsaw University of Technology, Poland, May 11, 2011.

D. Knees, A vanishing viscosity approach in damage mechanics, Workshop ``Variational Methods for Evolution'', December 5  10, 2011, Mathematisches Forschungsinstitut Oberwolfach, December 5, 2011.

D. Knees, Analysis und Numerik für quasistatische Rissausbreitung, Lectures in Continuum Mechanics, Universität Kassel, Fachbereich für Mathematik und Naturwissenschaften, December 12, 2011.

A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7  12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

J. Sprekels, Phase field models and hysteresis operators, Trends in Thermodynamics and Materials Theory 2011, December 15  17, 2011, Technische Universität Berlin, December 16, 2011.

J. Sprekels, Wellposedness, asymptotic behavior and optimal control of a nonstandard phase field model for diffusive phase segregation, Workshop on Optimal Control of Partial Differential Equations, November 28  December 1, 2011, Wasserschloss Klaffenbach, Chemnitz, November 30, 2011.

M. Liero, Rateindependent Kurzweil processes, Workshop ``Rateindependent Systems: Modeling, Analysis, and Computations'', August 30  September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 3, 2010.

A. Petrov, On a 3D model for shapememory alloys, Workshop ``Rateindependent Systems: Modeling, Analysis, and Computations'', August 30  September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 2, 2010.

A. Petrov, Viscoelastodynamic problem with Signorini boundary conditions, 81th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2010), Session on Applied Analysis, March 22  26, 2010, Universität Karlsruhe, March 25, 2010.

A. Glitzky, Existence of bounded steady state solutions to spinpolarized driftdiffusion systems, Workshop on Drift Diffusion Systems and Related Problems: Analysis, Algorithms and Computations, WIAS, Research Group ``Numerical Mathematics and Scientific Computing'', March 25, 2010.

A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Special Session on Reaction Diffusion Systems, May 25  28, 2010, Technische Universität Dresden, May 26, 2010.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified GibbsThomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22  26, 2010, Karlsruhe, March 23, 2010.

CH. Kraus, Inhomogeneous and anisotropic phasefield quantities in the sharp interface limit, 6th Singular Days 2010, April 29  May 1, 2010, WIAS, Berlin, April 30, 2010.

A. Mielke, Approaches to finitestrain elastoplasticity, SIAM Conference on Mathematical Aspects of Materials Science (MS10), May 23  26, 2010, Philadelphia, USA, May 23, 2010.

A. Mielke, Geometrische Nichtlinearitäten und LieGruppen in der Elastoplastizität, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, January 27, 2010.

A. Mielke, Gradient structures for electroreactiondiffusion systems with applications in photovoltaics, First Interdisciplinary Workshop of the GermanRussian Interdisciplinary Science Center (GRISC) ``Structure and Dynamics of Matter'', October 18  20, 2010, Freie Universität Berlin and HelmholtzZentrum Berlin für Materialien und Energie, October 19, 2010.

A. Mielke, Gradient structures for reactiondiffusion systems and semiconductor models with interface dynamics, International Conference on Evolution Equations, October 11  14, 2010, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, October 12, 2010.

J. Rehberg, On parabolic equations with nonhomogeneous Neumann boundary conditions, International Conference on Evolution Equations, October 11  14, 2010, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, October 11, 2010.

J. Rehberg, Optimal elliptic regularity near 3dimensional, heterogeneous Neumann vertices, 6th Singular Days on Asymptotic Methods for PDEs, April 29  May 1, 2010, WIAS, April 29, 2010.

J. Sprekels, Technical and mathematical problems in the Czochralski growth of single crystals, Workshop ``New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries'', January 31  February 6, 2010, Mathematisches Forschungsinstitut Oberwolfach, February 1, 2010.

H. Stephan, Asymptotisches Verhalten positiver Halbgruppen, Oberseminar Analysis, Technische Universität Dresden, Institut für Analysis, January 14, 2010.

H. Stephan, Evolution equations conserving positivity, Colloquium of Centre for Analysis, Scientific Computing and Applications (CASA), Technische Universiteit Einhoven, Netherlands, April 21, 2010.

J.A. Griepentrog, Maximal regularity for nonsmooth parabolic boundary value problems in SobolevMorrey spaces, International Conference on Elliptic and Parabolic Equations, November 30  December 4, 2009, WIAS, December 1, 2009.

S. Heinz, A model for the evolution of laminates, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Young Researchers Minisymposium ``Mathematics and Mechanics of Microstructure Evolution in Finite Plasticity'', February 9  13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

S. Heinz, The evolution of laminates, 8th GAMM Seminar on Microstructures, January 15  17, 2009, Universität Regensburg, NWFI Mathematik, January 17, 2009.

A. Petrov, On existence for viscoelastodymanic problems with unilateral boundary conditions, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Session ``Applied analysis'', February 9  13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

A. Petrov, On the error estimates for spacetime discretizations of rateindependent processes, 8th GAMM Seminar on Microstructures, January 15  17, 2009, Universität Regensburg, NWFI Mathematik, January 17, 2009.

A. Petrov, On the existence and error bounds for spacetime discretizations of a 3D model for shapememory alloys, Lisbon University, Center for Mathematics and Fundamental Applications, Portugal, September 17, 2009.

A. Petrov, On the numerical approximation of a viscoelastic problem with unilateral constrains, 7th EUROMECH Solid Mechanics Conference (ESMC2009), Minisymposium on Contact Mechanics, September 7  11, 2009, Instituto Superior Técnico, Lisbon, Portugal, September 8, 2009.

W. Dreyer, Stochastic evolution of many particle storage systems, 9th Hirschegg Workshop on Conservation Laws, September 6  12, 2009, Hirschegg, September 10, 2009.

J. Fuhrmann, Mathematical and numerical modeling of fuel cells and electrochemical flow cells, FraunhoferInstitut für Techno und Wirtschaftsmathematik, Kaiserslautern, December 1, 2009.

J. Fuhrmann, Modeling of reaction diffusion problems in semiconductor photoresists, Seventh Negev Applied Mathematical Workshop, July 6  8, 2009, Ben Gurion University of the Negev, Jacob Blaustein Institute for Desert Research, Sede Boqer Campus, Israel, July 8, 2009.

A. Glitzky, Discrete SobolevPoincaré inequalities for finite volume Voronoi approximations, Annual Meeting of the Deutsche MathematikerVereinigung and 17th Congress of the Österreichische Mathematische Gesellschaft, Section ``Partial Differential Equations'', September 20  25, 2009, Technische Universität Graz, Austria, September 21, 2009.

A. Glitzky, Discrete SobolevPoincaré inequalities using the $W^1,p$ seminorm in the setting of Voronoi finite volume approximations, International Conference on Elliptic and Parabolic Equations, November 30  December 4, 2009, WIAS, December 3, 2009.

H.Chr. Kaiser, Transient KohnSham theory, Jubiläumssymposium ``Licht  Materialien  Modelle'' (100 Jahre Innovation aus Adlershof), BerlinAdlershof, September 7  8, 2009.

H. Neidhardt, Linear nonautonomous Cauchy problems and evolution semigroups, International Bogolyubov Conference ``Problems of Theoretical and Mathematical Physics'', August 21  27, 2009, Moscow State University/Joint Institute for Nuclear Research, Dubna, Russian Federation, August 24, 2009.

J. Polzehl, Sequential multiscale procedures for adaptive estimation, The 1st Institute of Mathematical Statistics Asia Pacific Rim Meeting, June 28  July 1, 2009, Seoul National University, Institute of Mathematical Statistics, Korea (Republic of), July 1, 2009.

J. Rehberg, Functional analytic properties of the quantum mechanical particle density operator, International Workshop on Quantum Systems and Semiconductor Devices: Analysis, Simulations, Applications, April 20  24, 2009, Peking University, School of Mathematical Sciences, Beijing, China, April 21, 2009.

J. Rehberg, Quasilinear parabolic equations in distribution spaces, International Conference on Nonlinear Parabolic Problems in Honor of Herbert Amann, May 10  16, 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland, May 12, 2009.

H. Stephan, Inequalities for Markov operators, Positivity VI (Sixth Edition of the International Conference on Positivity and its Applications), July 20  24, 2009, El Escorial, Madrid, Spain, July 24, 2009.

H. Stephan, Modeling of diffusion prozesses with hidden degrees of freedom, Workshop on Numerical Methods for Applications, November 5  6, 2009, Lanke, November 6, 2009.

A. Petrov, Error estimates for spacetime discretizations of a 3D model for shapememory materials, IUTAM Symposium ``Variational Concepts with Applications to the Mechanics of Materials'', September 22  26, 2008, RuhrUniversität Bochum, Lehrstuhl für allgemeine Mechanik, September 24, 2008.

A. Petrov, Existence and approximation for 3D model of thermally induced phase transformation in shapememory alloys, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), Session ``Material models in solids'', March 31  April 4, 2008, Universität Bremen, April 1, 2008.

A. Petrov, Some mathematical results for a model of thermallyinduced phase transformations in shapememory materials, sc MatheonICM Workshop on Free Boundaries and Materials Modeling, March 17  18, 2008, WIAS, March 18, 2008.

H.Chr. Kaiser, A driftdiffusion model for semiconductors with internal interfaces, Annual Meeting of the Deutsche MathematikerVereinigung 2008, Minisymposium ``Analysis of ReactionDiffusion Systems with Internal Interfaces'', September 15  19, 2008, FriedrichAlexanderUniversität ErlangenNürnberg, September 15, 2008.

H.Chr. Kaiser, A thermodynamic approach to transient KohnSham theory, 100th Statistical Mechanics Conference, December 13  18, 2008, Rutgers, The State University of New Jersey, New Brunswick, USA, December 16, 2008.

H.Chr. Kaiser, On driftdiffusion KohnSham theory, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), session ``Applied Analysis'', March 31  April 4, 2008, University of Bremen, April 1, 2008.

CH. Kraus, A phasefield model with anisotropic surface tension in the sharp interface limit, Second GAMMSeminar on Multiscale Material Modelling, July 10  12, 2008, Universität Stuttgart, Institut für Mechanik (Bauwesen), July 12, 2008.

CH. Kraus, Ein Phasenfeldmodell vom CahnHilliardTyp im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

CH. Kraus, Phase field models and corresponding GibbsThomson laws. Part II, SIMTECH Seminar Multiscale Modelling in Fluid Mechanics, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, November 5, 2008.

J. Rehberg, Hölder continuity for elliptic and parabolic problems, AnalysisTag, Technische Universität Darmstadt, Fachbereich Mathematik, November 27, 2008.

A. Petrov, On the convergence for kinetic variational inequality to quasistatic variational inequality with application to elasticplastic systems with hardening, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 17, 2007.

A. Petrov, Thermally driven phase transformation in shapememory alloys, Workshop ``Analysis and Numerics of RateIndependent Processes'', February 26  March 2, 2007, Mathematisches Forschungsinstitut Oberwolfach, February 27, 2007.

A. Glitzky, Energy estimates for reactiondiffusion processes of charged species, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 16, 2007.

H.Chr. Kaiser, A driftdiffusion model of transient KohnSham theory, First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, Special Session ``Mathematics of Large Quantum Systems'', July 31  August 3, 2007, University of Warsaw, Poland, August 3, 2007.

CH. Kraus, On jump conditions at phase interfaces, Oberseminar über Angewandte Mathematik, December 10  15, 2007, Universität Freiburg, Abteilung für Angewandte Mathematik, December 11, 2007.

J. Rehberg, An elliptic model problem including mixed boundary conditions and material heterogeneities, Fifth Singular Days, April 23  27, 2007, International Center for Mathematical Meetings, Luminy, France, April 26, 2007.

J. Rehberg, Maximal parabolic regularity on Sobolev spaces, The Eighteenth Crimean Autumn Mathematical SchoolSymposium (KROMSH2007), September 17  29, 2007, LaspiBatiliman, Ukraine, September 18, 2007.

F. Schmid, An evolution model in contact mechanics with dry friction, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 19, 2007.

A. Petrov, Mathematical result on the stability of elasticplastic systems with hardening, European Conference on Smart Systems, October 26  28, 2006, Researching Training Network "New Materials, Adaptive Systems and their Nonlinearities: Modelling, Control and Numerical Simulation" within the European Commission's 5th Framework Programme, Rome, Italy, October 27, 2006.

CH. Kraus, Equilibrium conditions for liquidvapor system in the sharp interface limit, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, July 18, 2006.

A. Glitzky, Energy estimates for electroreactiondiffusion systems with partly fast kinetics, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 27, 2006.

A. Glitzky, Energy models where the equations are defined on different domains, GAMM Annual Meeting 2006, March 27  31, 2006, Technische Universität Berlin, March 29, 2006.

CH. Kraus, Equilibria conditions in the sharp interface limit of the van der WaalsCahnHilliard phase model, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14  16, 2006, Levico, Italy, September 14, 2006.

CH. Kraus, The sharp interface limit of the van der WaalsCahnHilliard model, PolishGerman Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

H. Neidhardt, Classical solutions of van Roosbroeck's semiconductor equations, Conference ``Mathematical aspects of quantum transport in mesoscopic systems'', December 18  20, 2006, Université de Toulon et Centre de Physique Théorique, Marseille, France, December 18, 2006.

H. Neidhardt, Perturbation theory of semigroups and evolution equations, Jahrestagung der Deutschen MathematikerVereinigung 2006, September 18  21, 2006, Universität Bonn, September 20, 2006.

J. Rehberg, Existence and uniqueness for van Roosbroeck's system in Lebesque spaces, Conference ``Recent Advances in Nonlinear Partial Differential Equations and Applications'', Toledo, Spain, June 7  10, 2006.

J. Rehberg, Regularity for nonsmooth elliptic problems, Crimean Autumn Mathematical School, September 20  25, 2006, Vernadskiy Tavricheskiy National University, Laspi, Ukraine, September 21, 2006.

CH. Kraus, On the sharp limit of the Van der WaalsCahnHilliard model, WIAS Workshop ``Dynamic of Phase Transitions'', November 30  December 3, 2005, Berlin, December 2, 2005.

CH. Kraus, On the sharp limit of the Van der WaalsCahnHilliard model, Workshop ``MicroMacro Modeling and Simulation of LiquidVapor Flows'', November 16  18, 2005, Universität Freiburg, Mathematisches Institut, Kirchzarten, November 17, 2005.

CH. Kraus, Maximale Konvergenz in höheren Dimensionen, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, May 24, 2005.

J. Rehberg, Elliptische und parabolische Probleme aus Anwendungen, Kolloquium im Fachbereich Mathematik, Universität Darmstadt, May 18, 2005.

J. Rehberg, Existence, uniqeness and regularity for quasilinear parabolic systems, International Conference ``Nonlinear Partial Differential Equations'', September 17  24, 2005, Institute of Applied Mathematics and Mechanics Donetsk, Alushta, Ukraine, September 18, 2005.

J. Rehberg, H$^1,q$regularity for linear, elliptic boundary value problems, Regularity for nonlinear and linear PDEs in nonsmooth domains  Analysis, simulation and application, September 5  7, 2005, Universität Stuttgart, Deutsche Forschungsgemeinschaft (SFB 404), Hirschegg, Austria, September 6, 2005.

J. Rehberg, Regularität für elliptische Probleme mit unglatten Daten, Oberseminar Prof. Escher/Prof. Schrohe, Technische Universität Hannover, December 13, 2005.

J. Rehberg, Some analytical ideas concerning the quantumdriftdiffusion systems, Workshop ``Problèmes spectraux nonlinéaires et modèles de champs moyens'', April 4  8, 2005, Institut Henri Poincaré, Paris, France, April 5, 2005.

J. Rehberg, Analysis of macroscopic and quantum mechanical semiconductor models, International Visitor Program ``Nonlinear Parabolic Problems'', August 8  November 18, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, November 1, 2005.

J. Rehberg, Existence, uniqueness and regularity for quasilinear parabolic systems, Conference ``Nonlinear Parabolic Problems'', October 17  21, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, October 20, 2005.

J. Rehberg, Elliptische und parabolische Probleme mit unglatten Daten, Technische Universität Darmstadt, Fachbereich Mathematik, December 14, 2004.

J. Rehberg, Quasilinear parabolic equations in $L^p$, Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, June 28  30, 2004, Universität Zürich, Institut für Mathematik, Switzerland, June 29, 2004.

J. Rehberg, The twodimensional van Roosbroeck system has solutions in $L^p$, Workshop ``Advances in Mathematical Semiconductor Modelling: Devices and Circuits'', March 2  6, 2004, ChineseGerman Centre for Science Promotion, Beijing, China, March 5, 2004.

U. Bandelow, Thermodynamic designed energy model, 3rd Topical meeting on Numerical Simulation of Semiconductor Optoelectronic Devices (NUSOD'03), October 14  16, 2003, University of Tokyo, Japan, October 14, 2003.

H.Chr. Kaiser, Classical solutions of van Roosbroeck's equations with discontinuous coefficients and mixed boundary conditions on twodimensional space domains, 19th GAMM Seminar Leipzig on Highdimensional problems  Numerical treatment and applications, January 23  25, 2003, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, January 25, 2003.

J. Rehberg, A combined quantum mechanical and macroscopic model for semiconductors, Workshop on Multiscale problems in quantum mechanics and averaging techniques, December 11  12, 2003, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, December 12, 2003.

J. Rehberg, Solvability and regularity for parabolic equations with nonsmooth data, International Conference ``Nonlinear Partial Differential Equations'', September 15  21, 2003, National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics, Alushta, September 17, 2003.
External Preprints

A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Preprint no. arXiv:1510.06219, Cornell University Library, arXiv.org, 2015.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
D.A. Gomes, S. Patrizi, Obstacle meanfield game problem, Preprint no. arXiv:1410.6942, Cornell University Library, arXiv.org, 2014.
Abstract
In this paper, we introduce and study a firstorder meanfield game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and powerlike nonlinearities. Since the obstacle operator is not differentiable, the equations for firstorder mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions. 
S. Neukamm, A. Gloria, F. Otto, An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, 2013.
Abstract
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author. 
A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasistatic damage evolution, Preprint no. 28PV10/26/0, Istituto di Matematica Applicata e Technologie Informatiche, 2010.

A. Mielke, On an evolutionary model for complete damage based on energies and stresses, Preprint no. 29PV10/27/0, pp. 2332, in: Rateindependent evolutions and material modeling, T. Roubíček, U. Stefanelli (eds.), Istituto di Matematica Applicata e Technologie Informatiche, 2010.

D. Knees, Griffithformula and Jintegral for a crack in a powerlaw hardening material, Preprint no. 2005/12, Universität Stuttgart, SFB 404, 2005.

M. Lichtner, M. Radziunas, Well posedness and smooth dependence for a semilinear hyperbolic system with nonsmooth data, Preprint no. 174, DFG Research Center sc Matheon, Technische Universität Berlin, 2004.
Contact
Applications
 Dynamics of semiconductor lasers
 Modeling and simulation of semiconductor structures
 Modeling of phase separation and damage in modern materials
 Modeling of thin films and nano structures on substrates
 Modeling, Simulation and Optimization for Biomedical Applications
 Nonlinear material models, multifunctional materials and hysteresis in connection with elastoplastic processes
 Optical pulses in nonlinear media
 Phase transition and hysteresis in the context of storage problems
 Quantum models for semiconductors
 Simulation, optimization and optimal control of production processes
Projects/Grants
 Analysis for thermomechanical models with internal variables
 Effective models for materials and interfaces with multiple scales
 Electromechanical coupling for semiconductor devices
 Fault networks and scaling properties of deformation accumulation
 Fractal and stochastic homogenization using variational techniques
 Interface dynamics: Bridging stochastic and hydrodynamic descriptions
 Nonlinear Fracture Dynamics: Modeling, Analysis, Approximation, and Applications
 Nonlinear electrokinetics in anisotropic microfluids  analysis, simulation, and optimal control
 Reliability of efficient approximation schemes for material discontinuities described by functions of bounded variation
 Stochastic spatial coagulation particle processes