Partial Differential Equations

Publications

Monographs

• U.W. Pohl, A. Strittmatter, A. Schliwa, M. Lehmann, T. Niermann, T. Heindel, S. Reitzenstein, M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Stressor-induced site control of quantum dots for single-photon sources, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 53--90, (Chapter Published).
  Abstract
  The strain field of selectively oxidized A10x current apertures in an A1GaAs/GaAs mesa is utilized to define the nucleation site of InGaAs/GaAs quantum dots. A design is developed that allows for the self-aligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmission-electron holography yield excellent agreement with the calculated strain field. Single-dot spectroscopy of site-controlled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantum-dot growth in a defect-free matrix. Implementation of such dots in an electrically driven pin structure yields single-dot electroluminescence. Single-photon emission with excellent purity is provided for this device using a Hanbury Brown and Twiss setup. The injection efficiency of the initian pin design is affected by a substantial lateral current spreading close to the oxide aperture. Allpying 3D carier-transport simulation a ppn doping profile is developed achieving a substantial improvement of the current injection.

• S. Rodt, P.-I. Schneider, L. Zschiedrich, T. Heindel, S. Bounouar, M. Kantner, Th. Koprucki, U. Bandelow, S. Burger, S. Reitzenstein, Deterministic quantum devices for optical quantum communication, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 285--359, (Chapter Published).
  Abstract
  Photonic quantum technologies are based on the exchange of information via single photons. The information is typically encoded in the polarization of the photons and security is ensured intrinsically via principles of quantum mechanics such as the no-cloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantum-light sources. Such light sources with close to ideal optical and quantum optical properties can be realized by self-assembled semiconductor quantum dots. These high-quality nanocrystals are predestined single-photon emitters due to their quasi zero-dimensional carrier confinement. Still, the development of practical quantum-dot-based sources of single photons and entangled-photon pairs for applications in photonic quantum technology and especially for the quantum-repeater scheme is very demanding and requires highly advanced device concepts and deterministic fabrication technologies. This is mainly explained by their random position and emission energy as well as by the low photon-extraction efficiency in simple planar device configurations.

• M. Kantner, Th. Hóhne, Th. Koprucki, S. Burger, H.-J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Multi-dimensional modeling and simulation of semiconductor nanophotonic devices, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 241--283, (Chapter Published).
  Abstract
  Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonicdevices is an important tool in the development of future integrated light sources and quantumdevices. Simulations can guide important technological decisions by revealing performance bottle-necks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multi-dimensional numerical simulationsfor computer-aided design tasks requires sophisticated numerical methods and modeling tech-niques. We review recent advances in device-scale modeling of quantum dot based single-photonsources and laser diodes by self-consistently coupling the optical Maxwell equations with semi-classical carrier transport models using semi-classical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multi-dimensional geometries, we have developed a novel hp-adaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semi-conductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including vertical-cavity surface-emitting lasers, grating couplers and single-photonsources.

• H. Neidhardt, A. Stephan, V.A. Zagrebnov, Chapter 13: Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces, in: Analysis and Operator Theory, Th.M. Rassias , V.A. Zagrebnov , eds., 146 of Springer Optimization and Its Applications, Springer, Cham, 2019, pp. 271--299, (Chapter Published), DOI 10.1007/978-3-030-12661-2_13 .

Articles in Refereed Journals

• A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, Th. Koprucki, Numerical simulation of TEM images for In(Ga)As/GaAs quantum dots with various shapes, Optical and Quantum Electronics, (2020), pp. 257/1--257/11 (published online on 05.05.2020), DOI 10.20347/WIAS.PREPRINT.2682 .
  Abstract
  We present a mathematical model and a tool chain for the numerical simulation of TEM images of semiconductor quantum dots (QDs). This includes elasticity theory to obtain the strain profile coupled with the Darwin-Howie-Whelan equations, describing the propagation of the electron wave through the sample. We perform a simulation study on indium gallium arsenide QDs with different shapes and compare the resulting TEM images to experimental ones. This tool chain can be applied to generate a database of simulated TEM images, which is a key element of a novel concept for model-based geometry reconstruction of semiconductor QDs, involving machine learning techniques.

• W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson 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. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .
  Abstract
  We consider an improved Nernst--Planck--Poisson 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 convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time 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 cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 2179--2197, 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 parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type 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.

• M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, SIAM Journal on Control and Optimization, 58 (2020), pp. 2288--2311, DOI 10.1137/19M1269944 .
  Abstract
  In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

• M. Heida, R. Kornhuber, J. Podlesny, Fractal homogenization of a multiscale interface problem, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 18 (2020), pp. 294--314, DOI 10.1137/18M1204759 .
  Abstract
  Inspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergence of the approximate problems is close to optimal.

• O. Marquardt, M.A. Caro, Th. Koprucki, P. Mathé, M. Willatzen, Multiband k $cdot$ p model and fitting scheme for ab initio-based electronic structure parameters for wurtzite GaAs, Phys. Rev. B., 101 (2020), pp. 235147/1--235147/12, DOI 10.1103/PhysRevB.101.235147 .
  Abstract
  We develop a 16-band k · p model for the description of wurtzite GaAs, together with a novel scheme to determine electronic structure parameters for multiband k · p models. Our approach uses low-discrepancy sequences to fit k · p band structures beyond the eight-band scheme to most recent ab initio data, obtained within the framework for hybrid-functional density functional theory with a screened-exchange hybrid functional. We report structural parameters, elastic constants, band structures along high-symmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttinger-like parameters, and optical matrix parameters) for a ten-band and a sixteen-band k · p model for wurtzite GaAs. Our fitting scheme can assign priorities to both selected bands and k points that are of particular interest for specific applications. Finally, ellipticity conditions can be taken into account within our fitting scheme in order to make the resulting parameter sets robust against spurious solutions.

• R. Rossi, U. Stefanelli, M. Thomas, Rate-independent evolution of sets, Discrete and Continuous Dynamical Systems -- Series S, published online in March 2020, DOI 10.3934/dcdss.2020304 .
  Abstract
  The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent 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 time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous 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, Asymptotic analysis of a tumor growth model with fractional operators, Asymptotic Analysis, 120 (2020), pp. 41--72, DOI 10.20347/WIAS.PREPRINT.2625 .
  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 Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich 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, well-posedness 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 well-posedness 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.

• D.H. Doan, A. Fischer, J. Fuhrmann, A. Glitzky, M. Liero, Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Journal of Computational Electronics, 19 (2020), pp. 1164--1174, DOI 10.1007/s10825-020-01505-6 .
  Abstract
  We present an electrothermal drift-diffusion model for organic semiconductor devices with Gauss-Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter-Gummel scheme. Using path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, which were only recently observed experimentally.

• TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, Discrete and Continuous Dynamical Systems -- Series S, published online in May 2020, DOI 10.3934/dcdss.2020345 .
  Abstract
  The notion of Energy-Dissipation-Principle convergence (EDP-convergence) 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 Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence 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 Marcelin--de Donder kinetics.

• J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics, pp. 1/1--1/47 (published online on 06.11.2020), DOI 10.1007/s10955-020-02663-4 .
  Abstract
  We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence 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. 443--485, DOI 10.20347/WIAS.PREPRINT.2576 .
  Abstract
  It is well-known 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 time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type 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 obstacle-type 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 non-autonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83--135, DOI 10.4171/PRIMS/56-1-5 .
  Abstract
  In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) 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 X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP 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 operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

• O. Souček, M. Heida, J. Málek, On a thermodynamic framework for developing boundary conditions for Korteweg fluids, International Journal of Engineering Science, 154 (2020), pp. 103316/1--28, DOI 10.1016/j.ijengsci.2020.103316 .
  Abstract
  We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.

• M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 22.07.2020, DOI 10.1002/zamm.201900288 .
  Abstract
  Phase-field 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 phase-field 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 phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green 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 time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

• M. Kantner, Th. Koprucki, Beyond just ``flattening the curve'': Optimal control of epidemics with purely non-pharmaceutical interventions, Journal of Mathematics in Industry, 10 (2020), published online on 18.08.2020, DOI 10.1186/s13362-020-00091-3 .
  Abstract
  When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple ?flattening of the curve?. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

• A. Mielke, A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 1765--1807, DOI 10.1142/S0218202520500360 .
  Abstract
  We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

• A. Mielke, T. Roubíček, Thermoviscoelasticity in Kelvin--Voigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), published online on 15.06.2020, DOI 10.1007/s00205-020-01537-z .
  Abstract
  The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent 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 gradient-dependent and interfacial recombination, Mathematical Models & Methods in Applied Sciences, 29 (2019), pp. 1819--1851, DOI 10.1142/S0218202519500350 .
  Abstract
  We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth 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 charge-carrier 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 divergence-form operators.

• D.H. Doan, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for organic semiconductor devices, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 55/1--55/18, DOI 10.1007/s00033-019-1089-z .
  Abstract
  We discuss drift-diffusion models for charge-carrier transport in organic semiconductor devices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The former leads to so-called Gauss-Fermi statistics, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlinear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift-diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauder's fixed-point theorem.

• M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptotic Analysis, 112 (2019), pp. 185--212, DOI 10.3233/ASY-181502 .
  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 rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

• G. Nika, B. Vernescu, Multiscale modeling of magnetorheological suspensions, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 14/1--14/19 (published online on 23.12.2019), DOI 10.1007/s00033-019-1238-4 .
  Abstract
  We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flowprofile resembles a Bingham fluid flow.

• 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. 9275--9284, DOI 10.1073/pnas.1820487116 .

• A. Stephan, H. Stephan, Memory equations as reduced Markov processes, Discrete and Continuous Dynamical Systems, 39 (2019), pp. 2133--2155, DOI 10.3934/dcds.2019089 .
  Abstract
  A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

• F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .
  Abstract
  We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

• F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/1--8/39 (published online on 28.11.2019), DOI 10.1007/s00526-019-1663-4 .
  Abstract
  We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

• Y. Zheng, A. Fischer, M. Sawatzki, D.H. Doan, M. Liero, A. Glitzky, S. Reineke, S.C.B. Mannsfeld, Introducing pinMOS memory: A novel, non-volatile organic memory device, Advanced Functional Materials, 30 (2020), pp. 1907119/1--1907119/10 (published online on 07.11.2019), DOI 10.1002/adfm.201907119 .
  Abstract
  In recent decades, organic memory devices have been researched intensely and they can, among other application scenarios, play an important role in the vision of an internet of things. Most studies concentrate on storing charges in electronic traps or nanoparticles while memory types where the information is stored in the local charge up of an integrated capacitance and presented by capacitance received far less attention. Here, a new type of programmable organic capacitive memory called p-i-n-metal-oxide-semiconductor (pinMOS) memory is demonstrated with the possibility to store multiple states. Another attractive property is that this simple, diode-based pinMOS memory can be written as well as read electrically and optically. The pinMOS memory device shows excellent repeatability, an endurance of more than 104 write-read-eraseread cycles, and currently already over 24 h retention time. The working mechanism of the pinMOS memory under dynamic and steady-state operations is investigated to identify further optimization steps. The results reveal that the pinMOS memory principle is promising as a reliable capacitive memory device for future applications in electronic and photonic circuits like in neuromorphic computing or visual memory systems.

• L. Adam, M. Hintermüller, D. Peschka, Th.M. Surowiec, Optimization of a multiphysics problem in semiconductor laser design, SIAM Journal on Applied Mathematics, 79 (2019), pp. 257--283, DOI 10.1137/18M1179183 .
  Abstract
  A multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system.

• F. Agnelli, A. Constantinescu, G. Nika, Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson's ratio, Continuum Mechanics and Thermodynamics, 32 (2020), pp. 433--449 (published online on 20.11.2019), DOI 10.1007/s00161-019-00851-6 .
  Abstract
  This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson's ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson's ratio below -1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

• A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary Gamma-convergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479--522, DOI 10.1007/s00028-019-00484-x .
  Abstract
  We consider the initial-value 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 semi-implicit 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. 792/1--792/32, 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 Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24.) 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), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order 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 Cahn--Hilliard systems with fractional operators and double obstacle potentials, Discrete and Continuous Dynamical Systems -- Series S, published online on 21.12.2019, urlhttps://doi.org/10.3934/dcdss.2020213, DOI 10.3934/dcdss.2020213 .
  Abstract
  In the recent paper ”Well-posedness and regularity for a generalized fractional Cahn--Hilliard system”, the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn--Hilliard 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 double-well potentials driving the phase separation process modeled by the Cahn--Hilliard 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 Cahn--Hilliard system” (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) 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 double-well potentials could be admitted. Results concerning existence of optimizers and first-order 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 control-to-state 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 so-called ”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. Farshbaf-Shaker and J. Sprekels in the paper ”A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles” (Appl. Math. Optim. 71 (2015), pp. 1-24) 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 Cahn--Hilliard 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 first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.

• P. Colli, G. Gilardi, J. Sprekels, Optimal velocity control of a convective Cahn--Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, Journal of Convex Analysis, 26 (2019), pp. 485--514.
  Abstract
  In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents a difficulty for the analysis. In contrast to the previous paper Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions by the same authors, the bulk and surface free energies are of double obstacle type, which renders the state constraint nondifferentiable. It is well known that for such cases standard constraint qualifications are not satisfied so that standard methods do not apply to yield the existence of Lagrange multipliers. In this paper, we overcome this difficulty by taking advantage of results established in the quoted paper for logarithmic nonlinearities, using a so-called `deep quench approximation'. We derive results concerning the existence of optimal controls and the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system.

• P. Colli, G. Gilardi, J. Sprekels, Recent results on well-posedness and optimal control for a class of generalized fractional Cahn--Hilliard systems, Control and Cybernetics, 48 (2019), pp. 153--197.

• P. Colli, G. Gilardi, J. Sprekels, Well-posedness and regularity for a fractional tumor growth model, Advances in Mathematical Sciences and Applications, 28 (2019), pp. 343--375.
  Abstract
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• P. Colli, G. Gilardi, J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn--Hilliard system, Rendiconti Lincei -- Matematica e Applicazioni, 30 (2019), pp. 437--478.
  Abstract
  In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn--Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable 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, self-adjoint 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 second-order elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general well-posedness and regularity results that extend corresponding results which are known for either the non-fractional 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 λ₁ of A plays an important und not entirely obvious role: if λ₁ is positive, then the operators A and B may be completely unrelated; if, however, λ₁ equals zero, then it must be simple and the corresponding one-dimensional 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, pp. published online on 21.10.2019, urlhttps://doi.org/10.1007/s00245-019-09618-6, DOI 10.1007/s00245-019-09618-6 .
  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 well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order 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 EDP-convergence, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 68/1--68/45, 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 Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. We call this notion relaxed EDP-convergence since the special structure of the dissipation functional may not be preserved under Gamma-convergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential.

• S.P. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels, Strong solutions to nonlocal 2D Cahn--Hilliard--Navier--Stokes systems with nonconstant viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), pp. 678--727, DOI 10.1088/1361-6544/aaedd0 .
  Abstract
  We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard 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 no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional 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) weak-strong 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 Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

• G. Gilardi, J. Sprekels, Asymptotic limits and optimal control for the Cahn--Hilliard system with convection and dynamic boundary conditions, Nonlinear Analysis. An International Mathematical Journal, 178 (2019), pp. 1--31, DOI 10.1016/j.na.2018.07.007 .
  Abstract
  In this paper, we study initial-boundary value problems for the Cahn--Hilliard 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 Cahn--Hilliard 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 first-order 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.

• L. Heltai, N. Rotundo, Error estimates in weighted Sobolev norms for finite element immersed interface methods, Computers & Mathematics with Applications. An International Journal, 78 (2019), pp. 3586--3604, DOI 10.1016/j.camwa.2019.05.029 .
  Abstract
  When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

• J. Lähnemann, M.O. Hill, J. Herranz, O. Marquardt, G. Gao, A. Al Hassan, A. Davtyan, S.O. Hruszkewycz, M.V. Holt, Ch. Huang, I. Calvo-Almazán, U. Jahn, U. Pietsch, L.J. Lauhon, L. Geelhaar, Correlated nanoscale analysis of the emission from wurtzite versus zincblende (In,Ga)As/GaAs nanowire core-shell quantum wells, ACS Nano, 19 (2019), pp. 4448--4457, DOI 10.1021/acs.nanolett.9b01241 .
  Abstract
  While the properties of wurtzite GaAs have been extensively studied during the past decade, little is known about the influence of the crystal polytype on ternary (In,Ga)As quantum well structures. We address this question with a unique combination of correlated, spatially resolved measurement techniques on core-shell nanowires that contain extended segments of both the zincblende and wurtzite polytypes. Cathodoluminescence hyperspectral imaging reveals a blue-shift of the quantum well emission energy by 75 ± 15 meV in the wurtzite polytype segment. Nanoprobe X-ray diffraction and atom probe tomography enable k•p calculations for the specific sample geometry to reveal two comparable contributions to this shift. First, there is a 30% drop in In mole fraction going from the zincblende to the wurtzite segment. Second, the quantum well is under compressive strain, which has a much stronger impact on the hole ground state in the wurtzite than in the zincblende segment. Our results highlight the role of the crystal structure in tuning the emission of (In,Ga)As quantum wells and pave the way to exploit the possibilities of three-dimensional band gap engineering in core-shell nanowire heterostructures. At the same time, we have demonstrated an advanced characterization toolkit for the investigation of semiconductor nanostructures.

• V. Laschos, A. Mielke, Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures, Journal of Functional Analysis, 276 (2019), pp. 3529--3576, 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 Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich 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 two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.

• P. Nestler, N. Schlömer, O. Klein, J. Sprekels, F. Tröltzsch, Optimal control of semiconductor melts by traveling magnetic fields, Vietnam Journal of Mathematics, 47 (2019), pp. 793--812, DOI 10.1007/s10013-019-00355-5 .
  Abstract
  In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely, one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat condution, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization.

• J. Sprekels, H. Wu, Optimal distributed control of a Cahn--Hilliard--Darcy system with mass sources, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 24.01.2019, urlhttps://doi.org/10.1007/s00245-019-09555-4, DOI 10.1007/s00245-019-09555-4 .
  Abstract
  In this paper, we study an optimal control problem for a two-dimensional Cahn--Hilliard--Darcy 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 control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order 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. 3--21, DOI 10.7900/jot.2017nov15.2233 .
  Abstract
  Under a mild regularity condition we prove that the generator of the interpolation of two C₀-semigroups is the interpolation of the two generators.

• A. Glitzky, M. Liero, Instationary drift-diffusion problems with Gauss--Fermi statistics and field-dependent mobility for organic semiconductor devices, Communications in Mathematical Sciences, 17 (2019), pp. 33--59, DOI 10.4310/cms.2019.v17.n1.a2 .
  Abstract
  This paper deals with the analysis of an instationary drift-diffusion model for organic semiconductor devices including Gauss--Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent energetic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength. To prove the existence of global weak solutions, we consider a problem with (for small densities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level ensure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densities. The estimates rely on Moser iteration techniques.

• P. Farrell, D. Peschka, Nonlinear diffusion, boundary layers and nonsmoothness: Analysis of challenges in drift-diffusion semiconductor simulations, Computers & Mathematics with Applications. An International Journal, 78 (2019), pp. 3731--3747, DOI 10.1016/j.camwa.2019.06.007 .
  Abstract
  We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

• M. Liero, S. Melchionna, The weighted energy-dissipation principle and evolutionary Gamma-convergence for doubly nonlinear problems, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 36/1--36/38, 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 so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence 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 Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven 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 λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application.

• M. Radziunas, J. Fuhrmann, A. Zeghuzi, H.-J. Wünsche, Th. Koprucki, C. Brée, H. Wenzel, U. Bandelow, Efficient coupling of dynamic electro-optical and heat-transport models for high-power broad-area semiconductor lasers, Optical and Quantum Electronics, 51 (2019), pp. 69/1--69/10, DOI 10.1007/s11082-019-1792-1 .
  Abstract
  In this work, we discuss the modeling of edge-emitting high-power broad-area semiconductor lasers. We demonstrate an efficient iterative coupling of a slow heat transport (HT) model defined on multiple vertical-lateral laser cross-sections with a fast dynamic electro-optical (EO) model determined on the longitudinal-lateral domain that is a projection of the device to the active region of the laser. Whereas the HT-solver calculates temperature and thermally-induced refractive index changes, the EO-solver exploits these distributions and provides time-averaged field intensities, quasi-Fermi potentials, and carrier densities. All these time-averaged distributions are used repetitively by the HT-solver for the generation of the heat sources entering the HT problem solved in the next iteration step.

Contributions to Collected Editions

• C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, On four numerical schemes for a unipolar degenerate drift-diffusion model, in: Proceedings of Finite Volumes for Complex Applications IX, Bergen, Norway, June 2020, R. Klöfkorn, F. Radu, E. Keijgavlen, J. Fuhrmann, eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 163--171, DOI 10.1007/978-3-030-43651-3_13 .

• U.W. Pohl, A. Strittmatter, A. Schliwa, M. Lehmann, T. Niermann, T. Heindel, S. Reitzenstein, M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Stressor-induced site control of quantum dots for single-photon sources, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Cham, 2020, pp. published online on 11.03.2020, DOI 10.1007/978-3-030-35656-9_3 .
  Abstract
  The strain field of selectively oxidized AlOx current apertures in an AlGaAs/GaAs mesa is utilized to define the nucleation site of InGaAs/GaAs quantum dots. A design is developed that allows for the self-aligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmission-electron holography yield excellent agreement with the calculated strain field. Single-dot spectroscopy of site-controlled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantum-dot growth in a defect-free matrix. Implementation of such dots in an electrically driven pin structure yields single-dot electroluminescence. Single-photon emission with excellent purity is proved for this device using a Hanbury Brown and Twiss setup. The injection efficiency of the initial pin design is affected by a substantial lateral current spreading close to the oxide aperture. Applying 3D carrier-transport simulation a ppn doping profile is developed achieving a substantial improvement of the current injection.

• S. Rodt, P.-I. Schneider, L. Zschiedrich, T. Heindel, S. Bounouar, M. Kantner, Th. Koprucki, U. Bandelow, S. Burger, S. Reitzenstein, Deterministic quantum devices for optical quantum communication, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Cham, 2020, pp. published online on 11.03.2020, DOI 10.1007/978-3-030-35656-9 .
  Abstract
  Photonic quantum technologies are based on the exchange of information via single photons. The information is typically encoded in the polarization of the photons and security is ensured intrinsically via principles of quantum mechanics such as the no-cloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantum-light sources. Such light sources with close to ideal optical and quantum optical properties can be realized by self-assembled semiconductor quantum dots. These high-quality nanocrystals are predestined single-photon emitters due to their quasi zero-dimensional carrier confinement. Still, the development of practical quantum-dot-based sources of single photons and entangled-photon pairs for applications in photonic quantum technology and especially for the quantum-repeater scheme is very demanding and requires highly advanced device concepts and deterministic fabrication technologies. This is mainly explained by their random position and emission energy as well as by the low photon-extraction efficiency in simple planar device configurations.

• S. Schulz, D. Chaudhuri, M. O'Donovan, S. Patra, T. Streckenbach, P. Farrell, O. Marquardt, Th. Koprucki, Multi-scale modeling of electronic, optical, and transport properties of III-N alloys and heterostructures, in: Proceedings Physics and Simulation of Optoelectronic Devices XXVIII, 11274, San Francisco, California, USA, 2020, pp. 416--426, DOI 10.1117/12.2551055 .

• J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, in: Proceedings of ``Finite Volumes for Complex Applications IX'', R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2020, pp. 625--633, DOI 10.1007/978-3-030-43651-3_59 .
  Abstract
  We discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss--Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

• M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.-J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Multi-dimensional modeling and simulation of semiconductor nanophotonic devices, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Cham, 2020, pp. 241--283, DOI 10.1007/978-3-030-35656-9_7 .
  Abstract
  Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semi-classical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperatures. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources.

• M. Kantner, Th. Koprucki, Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors, in: Proceedings of ``Finite Volumes for Complex Applications IX'', R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2020, pp. 173--182, DOI 10.1007/978-3-030-43651-3_14 .
  Abstract
  Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter--Gummel finite volume discretization for degenerate semiconductors obeying Fermi--Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

• M. Kantner, Th. Koprucki, Non-isothermal Scharfetter-Gummel scheme for electro-thermal transport simulation in degenerate semiconductors, in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2020, pp. 173--182, DOI 10.1007/978-3-030-43651-3_14 .
  Abstract
  Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter?Gummel finite volume discretization for degenerate semiconductors obeying Fermi?Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

• M. Kantner, A. Mielke, M. Mittnenzweig, N. Rotundo, Mathematical modeling of semiconductors: From quantum mechanics to devices, 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. 269--293, DOI 10.1007/978-3-030-33116-0 .
  Abstract
  We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of socalled GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

• M. Kantner, Th. Koprucki, H.-J. Wünsche, U. Bandelow, Simulation of quantum dot based single-photon sources using the Schrödinger--Poisson-Drift-Diffusion-Lindblad system, in: Proceedings of the 24th International Conference on Simulation of Semiconductor Processes and Devices (SISPAD 2019), F. Driussi, ed., 2019, pp. 355--358, DOI 10.1007/978-3-030-22116-0_11 .

• D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, 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. 295--318, DOI 10.1007/978-3-030-33116-0 .
  Abstract
  In this work we investigate a two-phase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a non-smooth two-homogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows.

• P. Colli, G. Gilardi, J. Sprekels, Nonlocal phase field models of viscous Cahn--Hilliard 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. 71--100, DOI 10.1007/978-3-030-33116-0 .
  Abstract
  A nonlocal phase field model of viscous Cahn--Hilliard type is considered. This model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli 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 well-posedness and stability of the system as well as on the distributed optimal control problem.

• TH. Koprucki, A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, J. Polzehl, On a database of simulated TEM images for In(Ga)As/GaAs quantum dots with various shapes, in: Proceedings of the 19th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2019, J. Piprek, K. Hinze, eds., IEEE Conference Publications Management Group, Piscataway, 2019, pp. 13--14, DOI 10.1109/NUSOD.2019.8807025 .

Preprints, Reports, Technical Reports

• M.H. Farshbaf Shaker, M. Thomas, Analysis of a compressible Stokes-flow 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 single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains.

• A. Mielke, M.A. Peletier, A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems with detailed balance, Preprint no. 2781, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2781 .
  Abstract, PDF (897 kByte)
  We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

• K.M. Gambaryan, O. Marquardt, T. Boeck, A. Trampert, Micro- and nano-scale engineering and structures shape architecture at nucleation from In-As-Sb-P composition liquid phase on an InAs(100) surface, Preprint no. 2775, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2775 .
  Abstract, PDF (4287 kByte)
  In this review paper we present results of the growth, characterization and electronic properties of In(As,Sb,P) composition strain-induced micro- and nanostructures. Nucleation is performed from In-As-Sb-P quaternary composition liquid phase in Stranski--Krastanow growth mode using steady-state liquid phase epitaxy. Growth features and the shape transformation of pyramidal islands, lens-shape and ellipsoidal type-II quantum dots (QDs), quantum rings and QD-molecules are under consideration. It is shown that the application of a quaternary In(As,Sb,P) composition wetting layer allows not only more flexible control of lattice-mismatch between the wetting layer and an InAs(100) substrate, but also opens up new possibilities for nanoscale engineering and nanoarchitecture of several types of nanostructures. HR-SEM, AFM, TEM and STM are used for nanostructure characterization. Optoelectronic properties of the grown structures are investigated by FTIR and photoresponse spectra measurements. Using an eight-band $mathbfkcdotmathbfp$ model taking strain and built-in electrostatic potentials into account, the electronic properties of a wide range of InAs$_1-x-y$Sb$_x$P$_y$ QDs and QD-molecules are computed. Two types of QDs mid-infrared photodetectors are fabricated and investigated. It is shown that the incorporation of QDs allows to improve some output device characteristics, in particularly sensitivity, and to broaden the spectral range.

• O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, Preprint no. 2773, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2773 .
  Abstract, PDF (706 kByte)
  The eight-band $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this well-established formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eight-band model as well as more sophisticated approaches for novel material systems and heterostructures.

• A. Mielke, Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials, Preprint no. 2771, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2771 .
  Abstract, PDF (342 kByte)
  We consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent 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 solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation 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 constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

• P. Colli, A. Signori, J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, Preprint no. 2770, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2770 .
  Abstract, PDF (451 kByte)
  This paper concerns a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness 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 first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order 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 second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

• A. Glitzky, M. Liero, G. Nika, An effective bulk-surface thermistor model for large-area organic light-emitting 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 large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear 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 current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used.

• D. Bothe, P.-É. Druet, On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell--Stefan and the Fick--Onsager approach, Preprint no. 2749, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2749 .
  Abstract, PDF (439 kByte)
  This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick--Onsager multicomponent diffusion fluxes or to the generalized Maxwell--Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell--Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell--Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes.

• M. Kantner, Th. Koprucki, Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions, Preprint no. 2748, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2748 .
  Abstract, PDF (3116 kByte)
  When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

• D. Peschka, M. Rosenau, Two-phase flows for sedimentation of suspensions, Preprint no. 2743, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2743 .
  Abstract, PDF (12 MByte)
  We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation.

• M. Heida, Stochastic homogenization on randomly perforated domains, Preprint no. 2742, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2742 .
  Abstract, PDF (1175 kByte)
  We study the existence of uniformly bounded extension and trace operators for W^1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local (δ, M)-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ”mesoscopic” connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on R^d and Ω in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.

• P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators, Preprint no. 2741, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2741 .
  Abstract, PDF (311 kByte)
  In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well 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 square-integrable functions on a bounded and smooth three-dimensional 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, M.H. Farshbaf Shaker, K. Shirakawa, N. Yamazaki, Optimal control for shape memory alloys of the one-dimensional Frémond model, Preprint no. 2737, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2737 .
  Abstract, PDF (700 kByte)
  In this paper, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential 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.

• TH. Eiter, On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations, Preprint no. 2727, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2727 .
  Abstract, PDF (258 kByte)
  The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent 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.

• 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 “Well-posedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93--128), 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 control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.

• R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in L^p, Preprint no. 2723, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2723 .
  Abstract, PDF (247 kByte)
  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.

• J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, Preprint no. 2721, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2721 .
  Abstract, PDF (326 kByte)
  In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion 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 extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well 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 L¹-norm. For such problems, we derive first-order 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 control-to-state 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 second-order sufficient optimality conditions.

• D. Bothe, P.-É. Druet, Well-posedness analysis of multicomponent incompressible flow models, Preprint no. 2720, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2720 .
  Abstract, PDF (465 kByte)
  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 Navier--Stokes 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 local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

• A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Preprint no. 2719, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2719 .
  Abstract, PDF (328 kByte)
  An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional 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 current-flow 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 three-dimensional 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 current-flow equation is given by two semilinear equations in the two-dimensional cross-sections 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.

• A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Preprint no. 2718, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2718 .
  Abstract, PDF (312 kByte)
  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 water-air 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 fractionally-damped 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 energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.

• E. Ipocoana, A. Zafferi, Further regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation, Preprint no. 2716, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2716 .
  Abstract, PDF (270 kByte)
  The aim of this paper is to establish new regularity results for a non-isothermal Cahn--Hilliard system in the two-dimensional 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.

• S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rate-independent 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 time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth 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 rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step 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 BV-functions, 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.

• H. Meinlschmidt, J. Rehberg, Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations, Preprint no. 2705, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2705 .
  Abstract, PDF (351 kByte)
  In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X^s-1,q_D(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

• P.-É. Druet, A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion, Preprint no. 2700, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2700 .
  Abstract, PDF (363 kByte)
  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 parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins 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 Maxwell-Stefan 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 Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

• M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck 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 Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root 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 non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

• G. Nika, B. Vernescu, Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids, Preprint no. 2673, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2673 .
  Abstract, PDF (1608 kByte)
  We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

• A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energy-dissipation landscapes via tilting -- Three types of EDP convergence, Preprint no. 2668, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2668 .
  Abstract, PDF (372 kByte)
  This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

• 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 non-symmetric second-order 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.

• 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 electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion 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.

• A. Stephan, Combinatorial considerations on the invariant measure of a stochastic matrix, Preprint no. 2627, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2627 .
  Abstract, PDF (225 kByte)
  The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.

• M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, Preprint no. 2626, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2626 .
  Abstract, PDF (424 kByte)
  In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

• K.M. Gambaryan, T. Boeck, A. Trampert, O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded composition quantum dots grown on InAs(100) substrate, Preprint no. 2623, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2623 .
  Abstract, PDF (745 kByte)
  We provide a detailed study of nucleation process, characterization, electronic and optical properties of graded composition quantum dots (GCQDs) grown from In-As-Sb-P composition liquid phase on an InAs(100) substrate in the Stranski-Krastanov growth mode. Our GCQDs exhibit diameters from 10 to 120 nm and heights from 2 to 20 nm with segregation profiles having a maximum Sb content of approximately 20% at the top and a maximum P content of approximately 15% at the bottom of the GCQDs so that hole confinement is expected in the upper parts of the GCQDs. Using an eight-band k · p model taking strain and built-in electrostatic potentials into account, we have computed the hole ground state energies and charge densities for a wide range of InAs_1-x-ySb_xP_y GCQDs as close as possible to the systems observed in experiment. Finally, we have obtained an absorption spectrum for an ensemble of GCQDs by combining data from both experiment and theory. Excellent agreement between measured and simulated absorption spectra indicates that such GCQDs can be grown following a theory-guided design for application in specific devices.

• G. Alì, N. Rotundo, Existence and uniqueness of solution for multidimensional parabolic PDAEs arising in semiconductor modeling, Preprint no. 2607, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2607 .
  Abstract, PDF (315 kByte)
  This paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with multi-dimensional parabolic-elliptic boundary value problems for the devices. We prove an existence and uniqueness result, and the asymptotic behavior of this mixed initial boundary value problem of partial differential-algebraic equations.

• B. Franchi, M. Heida, S. Lorenzani, A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation, Preprint no. 2595, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2595 .
  Abstract, PDF (452 kByte)
  In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.

• M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambda-convex gradient flows, Preprint no. 2594, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2594 .
  Abstract, PDF (429 kByte)
  In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.

• A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, Preprint no. 2593, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2593 .
  Abstract, PDF (387 kByte)
  This work is concerned with the analysis of a drift-diffusion 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 Gauss--Fermi 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 self-heating 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 drift-diffusion model for organic semiconductors on a sound analytical basis.

• A.F.M. TER Elst, R. Haller-Dintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients, Preprint no. 2590, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2590 .
  Abstract, PDF (383 kByte)
  Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order 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.

• P. Colli, G. Gilardi, J. Sprekels, Longtime behavior for a generalized Cahn--Hilliard system with fractional operators, Preprint no. 2588, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2588 .
  Abstract, PDF (248 kByte)
  In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit 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 Omega-limit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omega-limit 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.

• 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

• O. Marquardt, Semiconductor nanostructures, IKZ-WIAS Workshop (online), October 30, 2020, WIAS Berlin, IKZ Berlin, October 30, 2020.

• O. Marquardt, Data-driven electronic structure calculations for semiconductor nanostructures, Efficient algorithms for numerical problems - Workshop on the occasion of the retirement of Peter Mathé, January 17, 2020, WIAS Berlin, January 17, 2020.

• O. Marquardt, Electronic properties of semiconductor heterostructures using SPHInX, 20th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2020) - online event, software tutorial, September 14 - 25, 2020, Politecnico di Torino, September 14, 2020.

• O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded-composition quantum dots, 20th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2020) - online, September 14 - 25, 2020, Politecnico di Torino, September 14, 2020.

• O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded-composition quantum dots, CMD2020GEFES (online event), August 31 - September 4, 2020, European Physical Society & La Real Sociedad Española de Física, September 4, 2020.

• G. Nika, An existence result for a class of electrothermal drift-diffusion models with Fermi--Gauss statistics for organic semiconductors, Joint Mathematics Meeting, January 15 - 18, 2020, American Mathematical Society/ Mathematical Association of America, Denver, USA, January 15, 2020.

• P. Pelech, Separately global solutions to rate-independent systems: Applications to large-strain deformations of damageable solids, Thematic Einstein Semester: Kick-off Conference (online), October 26 - 30, 2020, WIAS Berlin, October 29, 2020.

• P. Pelech, Separately global solutions to rate-independent systems: Applications to large-strain deformations of damageable solids, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020.

• D. Peschka, Mathematical modeling and simulation of flows and the interaction with a substrate using energetic variational methods, Vortrag im Rahmen des SFB1194, Technische Universität Darmstadt, January 22, 2020.

• D. Peschka, Modeling and simulation of wetting and dewetting with dynamic contact angles, Wetting Dynamics 2020, Bonn, September 28 - 30, 2020.

• D. Peschka, L. Heltai, Fluid-structure interaction, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin.

• D. Peschka , Variational modeling of bulk and interface effects in fluid dynamics, SPP 2171 Advanced School ``Introduction to Wetting Dynamics'', February 17 - 21, 2020, Westfälische Wilhelms-Universität Münster, February 18, 2020.

• A. Stephan, Coarse-graining for gradient systems with applications to reaction systems, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 15, 2020.

• A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems, Annual Workshop of the GAMM Activity Group on Analysis of PDEs (online), September 30 - October 2, 2020, Universität Regensburg, October 1, 2020.

• A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Seminar ``Applied Analysis'', Eindhoven University of Technology, Centre for Analysis, Scientific Computing, and Applications - Mathematics and Computer Science, Netherlands, January 20, 2020.

• A. Stephan, EDP-convergence for nonlinear fast-slow reactions, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

• A. Stephan, On mathematical coarse-graining for linear reaction systems, 8th BMS Student Conference, February 19 - 21, 2020, Technische Universität Berlin, February 21, 2020.

• S. Tornquist, Dynamic phase-field fracture in visco-elastic materials, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 14, 2020.

• A. Zafferi, Lagrangian--Eulerian reduction of GENERIC systems, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 12, 2020.

• A. Glitzky, A hybrid model for the electrothermal behaviour of semiconductor devices, Annual Workshop of the GAMM Activity Group on Analysis of PDEs (online), September 30 - October 2, 2020, Universität Regensburg, October 1, 2020.

• K. Hopf, Global existence analysis of energy-reaction-diffusion systems, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 15, 2020.

• M. Thomas, Bulk-interface processes, Sitzung des Wissenschaftlichen Beirats, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, September 18, 2020.

• M. Thomas, Modeling and analysis of flows of concentrated suspensions (online talk), Kolloquium des Graduiertenkollegs, Universität Regensburg, July 10, 2020.

• M. Thomas, Modelling via energy and entropy functionals, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 14, 2020.

• M. Thomas, Nonlinear fraction dynamics: modeling, analysis, approximation, and applications, Vorstellung der Projektanträge im SPP 2256, Bad Honnef, January 30, 2020.

• TH. Eiter , Spatially asymptotic structure of time-periodic Navier--Stokes flows, Annual Workshop of the GAMM Activity Group on Analysis of PDEs (online), September 30 - October 2, 2020, Universität Regensburg, October 1, 2020.

• J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

• TH. Koprucki, The Mathematical Research Data Initiative (MaRDI) for the National Research Data Infrastructure, DMV Jahrestagung 2020 (online event), minisymposium on ``A Research Infrastructure Tailored for Mathematics in the Digital Age", September 14 - 17, 2020, Technische Universität Chemnitz, September 15, 2020.

• M. Liero, A. Mielke, Analysis for thermo-mechanical models with internal variables, Vorstellung der Projektanträge im SPP 2256, Bad Honnef, January 30, 2020.

• M. Liero, Evolutionary Gamma-convergence for multiscale problems, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 15, 2020.

• A. Mielke, Finite-strain 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 Gamma-convergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'', Julius-Maximilians-Universität Würzburg, June 5, 2020.

• A. Mielke, On finite-strain thermo-viscoelasticity, 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, Similarity solutions for Kolmogorov's two-equation model for turbulence, Workshop on Control of Self-Organizing Nonlinear Systems, September 2 - 3, 2020, Technische Universität Berlin, September 2, 2020.

• A. Mielke, Variational structures for the analysis of PDE systems, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 13, 2020.

• D. Peschka, Variational methods for viscoelastic flows and gelation, MATH+ Cluster Days 2020 (online event), September 21 - 22, 2020, Technische Universität Berlin, September 21, 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.

• A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, Microscopy Conference 2019, Poster session IM 4, Berlin, September 1 - 5, 2019.

• A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, BMS Summer School 2019: Mathematics of Deep Learning, Berlin, August 19 - 30, 2019.

• A. Maltsi, Model-based geometry reconstruction in TEM images, Sondierungsworkshop MPIE/WIAS ``Elektrochemie, Halbleiternanostrukturen und Metalle", October 14 - 15, 2019, Max-Planck-Institut für Eisenforschung GmbH Düsseldorf, October 15, 2019.

• A. Maltsi, Model-based geometry reconstruction in TEM images, Sondierungsworkshop MPIE/WIAS ``Elektrochemie, Halbleiternanostrukturen und Metalle", October 14 - November 15, 2019, Max-Planck-Institut für Eisenforschung GmbH Düsseldorf, October 15, 2019.

• A. Maltsi, Towards model-based geometry reconstruction of quantum dots from TEM, ``9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 19, 2019.

• 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 Fokker--Planck operator, ``9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 17, 2019.

• M. Heida, Stochastic homogenization of PDE on non-uniformly Lipschitz and percolating structures, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 24, 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 p-Laplacian 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 Fokker--Planck equation?, CRC1114 Colloquium, Freie Universität Berlin, SFB 1114, April 25, 2019.

• O. Marquardt, Charge confining mechanisms in III-V semiconductor nanowires, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2019), July 8 - 12, 2019, University of Ottawa, Canada, July 8, 2019.

• O. Marquardt, Data-driven electronic structure calculations for nanostructures (DESCANT), Sondierungsworkshop MPIE/WIAS ``Elektrochemie, Halbleiternanostrukturen und Metalle", October 14 - 15, 2019, Max-Planck-Institut für Eisenforschung GmbH Düsseldorf, October 15, 2019.

• O. Marquardt, Modelling the electronic properties of semiconductor nanowires, Engineering Physics Seminar, McMaster University, Hamilton, Canada, July 12, 2019.

• G. Nika, An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie.

• G. Nika, Homogenization for a multi-scale model of magnetorheological suspension, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Minisymposium MS ME-1-3 1 ``Emerging Problems in the Homogenization of Partial Differential Equations'', July 15 - 19, 2019, Valencia, Spain, July 15, 2019.

• D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films -- Asymptotic Approach vs. Gradient Dynamics, April 28 - May 3, 2019, Banff International Research Station for Mathematical Information and Discovery, Canada, April 30, 2019.

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

• D. Peschka, Gradient formulations with flow maps -- Mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, May 24, 2019.

• D. Peschka, Gradient structures for flows of concentrated suspensions - jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", February 18 - 22, 2019, Technische Universität Wien, Austria, February 20, 2019.

• D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flow, Begutachtungskolloquium für die Anträge des SPP 2171 ``Dynamische Benetzung flexibler, adaptiver und schaltbarer Oberflächen'', Mainz, February 7 - 8, 2019.

• D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.

• D. Peschka, Steering pattern formation of viscous flows, DMV-Jahrestagung 2019, Sektion ``Differentialgleichungen und Anwendungen'', September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 23, 2019.

• D. Peschka, ``Numerical methods for charge transport in semiconductors: FEM vs FV", 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

• A. Stephan, EDP-convergence for linear reaction diffusion systems with different time scales, Calculus of Variations on Schiermonnikoog 2019, July 1 - 5, 2019, Utrecht University, Schiermonnikoog, Netherlands, July 2, 2019.

• A. Stephan, EDP-convergence for linear reaction-diffusion systems with different time scales, Winter School ``Gradient Flows and Variational Methods in PDEs'', November 25 - 29, 2019, Universität Ulm, November 29, 2019.

• A. Stephan, Evolutionary Gamma-convergence for a linear reaction-diffusion system with different time scales, COPDESC-Workshop ``Calculus of Variation and Nonlinear Partial Differential Equations", March 25 - 28, 2019, Universität Regensburg, March 26, 2019.

• A. Stephan, Evolutionary Gamma-convergence for a linear reaction-diffusion system with different time scales, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 16, 2019.

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

• A. Stephan, Rigorous derivation of the effective equation of a linear reaction system with different time scales, 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.

• S. Tornquist, Variational problems involving Caccioppoli partitions, 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 19, 2019.

• A. Zafferi, An approach to multi-phase flows in geosciences, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, Turin, Italy, June 17 - 21, 2019.

• A. Zafferi, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, Zeuthen, May 20 - 22, 2019.

• A. Zafferi, Some regularity results for a non-isothermal Cahn-Hilliard 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.

• J. Sprekels, Optimal control of a Cahn--Hilliard--Darcy system with mass source modeling tumor growth, Università degli Studi di Pavia, Dipartimento di Matematica, Italy, May 14, 2019.

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

• A. Glitzky, Drift-diffusion problems with Gauss--Fermi statistics and field-dependent 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 Fokker--Planck 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 Fokker--Planck 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 gradient-regularized damage models, Mathematics Seminar Brescia, Università degli Studi di Brescia, Italy, March 13, 2019.

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

• M. Thomas, Analytical and numerical aspects for the approximation of gradient-regularized damage models, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS A3-2-26 ``Phase-Field Models in Simulation and Optimization'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

• M. Thomas, Analytical and numerical aspects of rate-independent gradient-regularized 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 rate-independent and rate-dependent systems, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, June 17 - 19, 2019, Politecnico di Torino, Turin, Italy.

• M. Thomas, Coupling of rate-independent and rate-dependent 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 rate-independent and rate-dependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.

• M. Thomas, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, May 20 - 22, 2019, Freie Universität Berlin, Zeuthen, May 21, 2019.

• M. Thomas, GENERIC structures with bulk-interface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

• M. Thomas, Gradient structures for flows of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME-7-75 ``Recent Advances in Understanding Suspensions and Granular Media Flow'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

• M. Thomas, Rate-independent 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.

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

• TH. Koprucki, Datenmanagement - Forschungsdaten in Modellierung und Simulation, Block-Seminar des SFB 787 ``Nanophotonik'', May 6 - 8, 2019, Technische Universität Berlin, Graal-Müritz, May 6, 2019.

• TH. Koprucki, Math4NFDI - a consortium for mathematics, Getting Ready: Leibniz in der Nationalen Forschungsdateninfrastruktur (NFDI), June 5, 2019, Leibniz Geschäftsstelle, Berlin, June 5, 2019.

• TH. Koprucki, Model pathway diagrams for the representation of mathematical models, DMV-Jahrestagung 2019, Mini-symposium ``FAIRmath: Opening mathematical research data for the next generation", September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 24, 2019.

• TH. Koprucki, Multi-dimensional modeling and simulation of semiconductor devices, Physikalisches Kolloquium, Technische Universität Chemnitz, Institut für Physik, November 27, 2019.

• TH. Koprucki, On a database of simulated TEM images for In(Ga)As/GaAs quantum dots with various shapes, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2019) , Session ``Nano-structures", July 8 - 12, 2019, University of Ottawa, Canada, July 8, 2019.

• TH. Koprucki, Towards multiscale modeling of III-N-based LEDs, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2019) , Session ``Postdeadline Session and Outlook", July 8 - 12, 2019, University of Ottawa, Canada, July 12, 2019.

• M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, 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, COPDESC-Workshop ``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 Gamma-convergence for gradient systems, Mathematisches Kolloquium, Albert-Ludwigs-Universität 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 Wilhelms-Universität Münster, September 17, 2019.

• A. Mielke, Gradient systems and evolutionary Gamma-convergence, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT -- Karlsruher Institut für Technologie, September 24, 2019.

• A. Mielke, On Kolmogorov's two-equation 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 initial-boundary 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, Thermodynamical modeling via GENERIC: From quantum mechanics to semiconductor devices, Institute of Thermomechanics's Seminar, Czech Academy of Sciences, Prague, March 21, 2019.

• A. Mielke, Variational methods in time-dependent material models with finite-strain 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 ME-1-3 9 ``Entropy Methods for Multi-dimensional 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, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, June 5, 2019.

• A. Mielke, Transport versus growth and decay: The (spherical) Hellinger--Kantorovich 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 Lax--Milgram 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, Well-posedness for the Keller--Segel 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. Tornquist, Towards the analysis of dynamic phase-field fracture, Spring School on Variational Analysis 2019, Paseky, Czech Republic, May 19 - 25, 2019.

• S. Tornquist, Towards the analysis of dynamic phase-field fracture, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, Turin, Italy, June 17 - 21, 2019.