Publications

Monographs

  • O. Marquardt, V.M. Kaganer, P. Corfdir, Chapter 12 ``Nanowires'' in Handbook of Optoelectronic Device Modeling and Simulations: Fundamentals, Materials, Nanostructures, LEDs, and Amplifiers, J. Piprek, ed., 1 of Series in Optics and Optoelectronics, CRC Press, 2017, pp. 395--415, (Chapter Published).

  • S. Jachalski, D. Peschka, S. Bommer, R. Seemann, B. Wagner, Chapter 18: Structure Formation in Thin Liquid--Liquid Films, in: Transport Processes at Fluidic Interfaces, D. Bothe, A. Reusken, eds., Advances in Mathematical Fluid Mechanics, Birkhäuser, Springer International Publishing AG, Cham, Switzerland, 2017, pp. 531--574, (Chapter Published), DOI 10.1007/978-3-319-56602-3 .
    Abstract
    We revisit the problem of a liquid polymer that dewets from another liquid polymer substrate with the focus on the direct comparison of results from mathematical modeling, rigorous analysis, numerical simulation and experimental investigations of rupture, dewetting dynamics and equilibrium patterns of a thin liquid-liquid system. The experimental system uses as a model system a thin polystyrene (PS) / polymethylmethacrylate (PMMA) bilayer of a few hundred nm. The polymer systems allow for in situ observation of the dewetting process by atomic force microscopy (AFM) and for a precise ex situ imaging of the liquid--liquid interface. In the present study, the molecular chain length of the used polymers is chosen such that the polymers can be considered as Newtonian liquids. However, by increasing the chain length, the rheological properties of the polymers can be also tuned to a viscoelastic flow behavior. The experimental results are compared with the predictions based on the thin film models. The system parameters like contact angle and surface tensions are determined from the experiments and used for a quantitative comparison. We obtain excellent agreement for transient drop shapes on their way towards equilibrium, as well as dewetting rim profiles and dewetting dynamics.

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

  • A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 187--249, (Chapter Published).
    Abstract
    In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions.
    We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

  • W. König, J. Sprekels, eds., Karl Weierstraß (1815--1897): Aspekte seines Lebens und Werkes -- Aspects of his Life and Work, Mathematik -- Analysis, Springer Spektrum, Wiesbaden, 2016, xv+289 pages, (Collection Published).
    Abstract
    Der Berliner Mathematiker Karl Weierstraß (1815-1897) lieferte grundlegende Beiträge zu den mathematischen Fachgebieten der Funktionentheorie, Algebra und Variationsrechnung. Er gilt weltweit als Begründer der mathematisch strengen Beweisführung in der Analysis. Mit seinem Namen verbunden ist zum Beispiel die berühmte Epsilon-Delta-Definition des Begriffs der Stetigkeit reeller Funktionen. Weierstraß? Vorlesungszyklus zur Analysis in Berlin wurde weithin gerühmt und er lehrte teilweise vor 250 Hörern aus ganz Europa; diese starke mathematische Schule prägt bis heute die Mathematik. Aus Anlass seines 200. Geburtstags am 31. Oktober 2015 haben internationale Experten der Mathematik und Mathematikgeschichte diesen Festband zusammengestellt, der einen Einblick in die Bedeutung von Weierstraß? Werk bis zur heutigen Zeit gibt. Die Herausgeber des Buches sind leitende Wissenschaftler am Weierstraß-Institut für Angewandte Analysis und Stochastik in Berlin, die Autoren eminente Mathematikhistoriker.

Articles in Refereed Journals

  • K. Disser, A.F.M. TER Elst, J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, Journal of Differential Equations, 262 (2017), pp. 2039--2072.
    Abstract
    We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

  • K. Disser, J. Rehberg, A.F.M. TER Elst, Hölder estimates for parabolic operators on domains with rough boundary, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XVII (2017), pp. 65--79.
    Abstract
    In this paper we investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove Hölder continuity of the solution in space and time.

  • S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Asymptotic Analysis, 105 (2017), pp. 3--26, DOI 10.20347/WIAS.PREPRINT.2354 .
    Abstract
    Based on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , ½] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.

  • M. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1303--1327.
    Abstract
    We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

  • M. Heida, Stochastic homogenization of rate-independent systems, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 853--894, DOI 10.1007/s00161-017-0564-z .
    Abstract
    We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

  • M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .
    Abstract
    Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

  • O. Marquardt, Th. Krause, V. Kaganer, J. Martin-Sánchez, M. Hanke, O. Brandt, Influence of strain relaxation in axial $In_xGa_1-xN/GaN$ nanowire heterostructures on their electronic properties, Nanotechnology, 28 (2017), pp. 215204/1--215204/6, DOI 10.1088/1361-6528/aa6b73 .
    Abstract
    We present a systematic theoretical study of the influence of elastic strain relaxation on the built-in electrostatic potentials and the electronic properties of axial InxGa1-xN/GaN nanowire (NW) heterostructures. Our simulations reveal that for a sufficiently large ratio between the thickness of the InxGa1-xN disk and the diameter of the NW, the elastic relaxation leads to a significant reduction of the built-in electrostatic potential in comparison to a planar system of similar layer thickness and In content. In this case, the ground state transition energies approach constant values with increasing thickness of the disk and only depend on the In content, a behavior usually associated to that of a quantum well free of built-in electrostatic potentials. We show that the structures under consideration are by no means field-free, and the built-in potentials continue to play an important role even for ultrathin NWs. In particular, strain and the resulting polarization potentials induce complex confinement features of electrons and holes, which depend on the In content, shape, and dimensions of the heterostructure.

  • O. Marquardt, M. Ramsteiner, P. Corfdir, L. Geelhaar, O. Brandt, Modeling the electronic properties of GaAs polytype nanostructures: Impact of strain on the conduction band character, Phys. Rev. B., 95 (2017), pp. 245309/1--245309/8, DOI 10.1103/PhysRevB.95.245309 .
    Abstract
    We study the electronic properties of GaAs nanowires composed of both the zinc-blende and wurtzite modifications using a ten-band k-p model. In the wurtzite phase, two energetically close conduction bands are of importance for the confinement and the energy levels of the electron ground state. These bands form two intersecting potential landscapes for electrons in zinc-blende/wurtzite nanostructures. The energy difference between the two bands depends sensitively on strain, such that even small strains can reverse the energy ordering of the two bands. This reversal may already be induced by the non-negligible lattice mismatch between the two crystal phases in polytype GaAs nanostructures, a fact that was ignored in previous studies of these structures. We present a systematic study of the influence of intrinsic and extrinsic strain on the electron ground state for both purely zinc-blende and wurtzite nanowires as well as for polytype superlattices. The coexistence of the two conduction bands and their opposite strain dependence results in complex electronic and optical properties of GaAs polytype nanostructures. In particular, both the energy and the polarization of the lowest intersubband transition depends on the relative fraction of the two crystal phases in the nanowire.

  • M. Mittnenzweig, A. Mielke, An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models, Journal of Statistical Physics, 167 (2017), pp. 205--233, DOI 10.1007/s10955-017-1756-4 .
    Abstract
    We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems.

  • H. Neidhardt, A. Stephan, V.A. Zagrebnov, On convergence rate estimates for apploximations of solutions operators of linear non-autonomous evolution equations, Nanosystems: Physics, Chemistry, Mathematics, 8 (2017), pp. 202--215.

  • R. Rossi, M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 1419--1494.
    Abstract
    We address the analysis of an abstract system coupling a rate-independet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

  • R. Rossi, M. Thomas, From adhesive to brittle delamination in visco-elastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 1489--1546, DOI 10.1142/S0218202517500257 .
    Abstract
    In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

  • J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, Journal of Functional Analysis, 273 (2017), pp. 1970--2025.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 2518--2546.
    Abstract
    In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn-Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 37--54.
    Abstract
    This paper is concerned with a phase field system of Cahn--Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates

  • P. Colli, G. Gilardi, J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evolution Equations and Control Theory, 6 (2017), pp. 35--58.
    Abstract
    This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called `deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.

  • P. Colli, G. Gilardi, J. Sprekels, Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 1732--1760, DOI 10.1137/16M1087539 .
    Abstract
    In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.

  • P. Colli, G. Gilardi, J. Sprekels, Recent results on the Cahn--Hilliard equation with dynamic boundary condition, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 10 (2017), pp. 5--21.

  • P. Exner, A.S. Kostenko, M.M. Malamud, H. Neidhardt, Infinite quantum graphs, Rossiiskaya Akademiya Nauk. Doklady Akademii Nauk, 472 (2017), pp. 253--258.

  • P. Exner, A.S. Kostenko, M.M. Malamud, H. Neidhardt, Rossiiskaya Akademiya Nauk. Doklady Akademii Nauk, 472 (2017), pp. 1--6.

  • M.M. Malamud, H. Neidhardt, H. Peller, A trace formula for functions of contractions and analytyc operator Lipschitz functions, Comptes Rendus Mathematique. Academie des Sciences. Paris, 355 (2017), pp. 806--811.

  • J. Sprekels, E. Valdinoci, A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM Journal on Control and Optimization, 55 (2017), pp. 70--93.
    Abstract
    In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the “control parameter” that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new classof identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter also the domain of definition, and thus the underlying function space, of the fractional operator changes.

  • A. Glitzky, M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 34 (2017), pp. 536--562.
    Abstract
    We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.

  • M. Thomas, Ch. Zanini, Cohesive zone-type delamination in visco-elasticity, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1487--1517, DOI 10.20347/WIAS.PREPRINT.2350 .
    Abstract
    We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

    Due to the presence of multivalued and unbounded operators featuring non-penetration and the `memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIAS-Preprint2113].

  • P. Farrell, Th. Koprucki, J. Fuhrmann, Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics, Journal of Computational Physics, 346 (2017), pp. 497--513, DOI 10.1016/j.jcp.2017.06.023 .
    Abstract
    For a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter--Gummel scheme to non-Boltzmann (e.g. Fermi--Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

  • M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Optical and Quantum Electronics, 330 (2017), DOI DOI 10.1007/s11082-017-1167-4 .
    Abstract
    Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.

  • A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities, Journal of Nonlinear Science, (2017), published online on 11.11.2017, DOI 10.1007/s00332-017-9427-9 .
    Abstract
    We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

  • A. Mielke, R. Rossi, G. Savaré, Global existence results for viscoplasticity at finite strain, Archive for Rational Mechanics and Analysis, (2017), published online on 20.9.2017, DOI 10.1007/s00205-017-1164-6 .
    Abstract

    We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable.
    The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.
  • A. Mielke, C. Patz, Uniform asymptotic expansions for the infinite harmonic chain, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 36 (2017), pp. 437--475, DOI 10.4171/ZAA/1596 .
    Abstract
    We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.20347/WIAS.PREPRINT.2165 .
    Abstract
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • K. Disser, G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid, Archive for Rational Mechanics and Analysis, 221 (2016), pp. 487--526.
    Abstract
    We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result.
    In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

  • N. Rotundo, T.-Y. Kim, W. Jiang, L. Heltai, E. Fried, Error estimates of B-spline based finite-element method for the wind-driven ocean circulation, Journal of Scientific Computing, 69 (2016), pp. 430--459.
    Abstract
    We present the error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale wind-driven ocean circulation. In particular, we derive optimal error estimates for h-refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and Stommel--Munk models. Numerical results on rectangular and embedded geometries confirm the error analysis.

  • S.P. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 1957--1993.
    Abstract
    We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

  • M. Heida, B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016), pp. 5--23.

  • M. Kantner, Th. Koprucki, Numerical simulation of carrier transport in semiconductor devices at cryogenic temperatures, Optical and Quantum Electronics, 48 (2016), pp. 543/1--543/7.
    Abstract
    At cryogenic temperatures the electron?hole plasma in semiconductors becomes strongly degenerate, leading to very sharp internal layers, extreme depletion in intrinsic domains and strong nonlinear diffusion. As a result, the numerical simulation of the drift?diffusion system suffers from serious convergence issues using standard methods. We consider a one-dimensional p-i-n diode to illustrate these problems and present a simple temperature-embedding scheme to enable the numerical simulation at cryogenic temperatures. The method is suitable for forward-biased devices as they appear e.g. in optoelectronic applications. Moreover, the method can be applied to wide band gap semiconductors where similar numerical issues occur already at room temperature.

  • M. Kantner, U. Bandelow, Th. Koprucki, J.-H. Schulze, A. Strittmatter, H.-J. Wünsche, Efficient current injection into single quantum dots through oxide-confined pn-diodes, IEEE Transactions on Electron Devices, 63 (2016), pp. 2036--2042.
    Abstract
    Current injection into single quantum dots embedded in vertical pn-diodes featuring oxide apertures is analyzed in the low-injection regime suitable for single-photon emitters. Experimental and theoretical evidence is found for a rapid lateral spreading of the carriers after passing the oxide aperture in the conventional pin-design. By an alternative design employing p-doping up to the oxide aperture the current spreading can be suppressed resulting in an enhanced current confinement and increased injection efficiencies, both, in the continuous wave and under pulsed excitation.

  • M. Liero, A. Mielke, G. Savaré, Optimal transport in competition with reaction: The Hellinger--Kantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 2869--2911.
    Abstract
    We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ d, which we call Hellinger-Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties.

  • H. Neidhardt, A. Stephan, V. Zagrebnov, On convergence rate estimates for approximations of solution operators for linear non-autonomous evolutions equations, Nanosystems: Physics, Chemistry, Mathematics, 8 (2017), DOI 10.17586/2220/8054-2017-8-2-tba .
    Abstract
    We improve some recent estimates of the rate of convergence for product approximations of solution operators for linear non-autonomous Cauchy problem. The Trotter product formula approximation is proved to converge to the solution operator-norm. We estimate the rate of convergence of this approximation. The result is applied to diffusion equation perturbed by a time-dependent potential.

  • D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, M. Virgilio, S. Guha, Th. Schröder, G. Cappellini, Th. Koprucki, Robustness analysis of a device concept for edge-emitting lasers based on strained germanium, Optical and Quantum Electronics, 48 (2016), pp. 156/1--156/7.
    Abstract
    We consider a device concept for edge-emitting lasers based on strained germanium microstrips. The device features an inhomogeneous tensile strain distribution generated by a SiN stressor deposited on top of the Ge microstrip. This geometry requires a lateral contact scheme and hence a full two-dimensional description. The two-dimensional simulations of the carrier transport and of the optical field, carried out in a cross section of the device orthogonal to the optical cavity, use microscopic calculations of the strained Ge material gain as an input. In this paper we study laser performance and robustness against Shockley-Read-Hall lifetime variations and device sensitivity to different strain distributions.

  • D. Peschka, N. Rotundo, M. Thomas, Towards doping optimization of semiconductor lasers, Journal of Computational and Theoretical Transport, 45 (2016), pp. 410--423.
    Abstract
    We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.

  • A. Boitsev, H. Neidhardt, I.Y. Popov, Dirac operator coupled to bosons, Nanosystems: Physics, Chemistry, Mathematics, 7 (2016), pp. 332--339.

  • M. Bulíček, A. Glitzky, M. Liero, Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 3496--3514.
    Abstract
    We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L1 term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the L1 right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.

  • P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016), pp. 195--225.
    Abstract
    A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

  • P. Colli, G. Gilardi, J. Sprekels, Constrained evolution for a quasilinear parabolic equation, Journal of Optimization Theory and Applications, 170 (2016), pp. 713--734.
    Abstract
    In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy--Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L2(Ω). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.

  • P. Colli, G. Gilardi, J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), pp. 246--281.
    Abstract
    We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The 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 model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

  • P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016), pp. 7940--7964.
    Abstract
    This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.

  • M. Erbar, M. Fathi, V. Laschos, A. Schlichting, Gradient flow structure for McKean--Vlasov equations on discrete spaces, Discrete and Continuous Dynamical Systems, 36 (2016), pp. 6799--6833.
    Abstract
    In this work, we show that a family of non-linear mean-field equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity

  • A. Fiebach, A. Glitzky, A. Linke, Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 141--174.
    Abstract
    We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.

  • R. Haller-Dintelmann, A. Jonsson, D. Knees, J. Rehberg, Elliptic and parabolic regularity for second order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016), pp. 5007--5026.
    Abstract
    We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces.

  • S. Heinz, A. Mielke, Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 374 (2016), pp. 20150171/1--20150171/23.
    Abstract
    We revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in A. Mielke, F. Theil, and V.I. Levita “A variational formulation of rate--independent phase transformations using an extremum principle", Arch. Rational Mech. Anal., 162, 137-177, 2002 ([MTL02]). This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable numerical approximations via space-time discretization, and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.

  • F. Kaschura, A. Fischer, M.P. Klinger, D.H. Doan, Th. Koprucki, A. Glitzky, D. Kasemann, J. Widmer, K. Leo, Operation mechanism of high performance organic permeable base transistors with an insulated and perforated base electrode, Journal of Applied Physics, 120 (2016), pp. 094501/1--094501/8.

  • H. Meinlschmidt, J. Rehberg, Hölder-estimates for non-autonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016), pp. 147--184.
    Abstract
    In this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.

  • S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Advances in Mathematical Sciences and Applications, 25 (2016), pp. 117--131.
    Abstract
    This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.

  • S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in two dimensions, SIAM Journal on Control and Optimization, 54 (2016), pp. 221 -- 250.
    Abstract
    We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

  • S.P. Frigeri, G.C. Gal, M. Grasselli, On nonlocal Cahn--Hilliard--Navier--Stokes systems in two dimensions, Journal of Nonlinear Science, 26 (2016), pp. 847--893.
    Abstract
    We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the strong-weak uniqueness in the case of viscosity depending on the order parameter, provided that the mobility is constant and the potential is regular. In the case of constant viscosity, on account of the uniqueness results we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor.

  • A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of Non-Equilibrium Thermodynamics, 41 (2016), pp. 141--149.
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • A. Mielke, T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 2203--2236.
    Abstract
    Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.
     
    In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

  • A. Mielke, R. Rossi, G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, Journal of the European Mathematical Society (JEMS), 18 (2016), pp. 2107--2165.
    Abstract
    Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for in nite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their di erentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on re ned lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

Contributions to Collected Editions

  • M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Efficient coupling of inhomogeneous current spreading and electro-optical models for simulation of dynamics in broad-area semiconductor lasers, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 217--218.

  • TH. Koprucki, M. Kohlhaase, D. Müller, K. Tabelow, Mathematical models as research data in numerical simulation of opto-electronic devices, in: Numerical Simulation of Optoelectronic Devices (NUSOD), 2017, pp. 225-- 226, DOI 10.1109/NUSOD.2017.8010073 .
    Abstract
    Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the stationary one-dimensional drift-diffusion equations (van Roosbroeck system).

  • M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, Modeling and simulation of electrothermal feedback in large-area organic leds, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 105--106, DOI 10.1109/NUSOD.2017.8010013 .

  • M. Bulíček, A. Glitzky, M. Liero, Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions, in: PDE 2015: Theory and Applications of Partial Differential Equations (PDE 2015), H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., 10 of Discrete and Continuous Dynamical Systems, Series S, American Institute of Mathematical Sciences, Springfield, 2017, pp. 697--713.
    Abstract
    We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

  • Y. Granovskyi, M.M. Malamud, H. Neidhardt, A. Posilicano, To the spectral theory of vector-valued Sturm--Liouville operators with summable potentials and point interactions, in: Functional Analysis and Operator Theory for Quantum Physics, 271--313 of EMS Congress Reports, Eur. Math. Soc, Zürich, 2017.

  • Y. Granovskyi, M.M. Malamud, H. Neidhardt, A. Posilicano, To the spectral theory of vector-valued Sturm--Liouville operators with summable potentials and point interactions, in: Functional Analysis and Operator Theory for Quantum Physics, J. Dittrich, H. Kovarik, A. Laptev, eds., EMS Series of Congress Reports, EMS Publishing House, 2017, pp. 271--313.

  • M. Kohlhase, Th. Koprucki, D. Müller, K. Tabelow, Mathematical models as research data via flexiformal theory graphs, in: Intelligent Computer Mathematics: 10th International Conference, CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceedings, H. Geuvers, M. England, O. Hasan, F. Rabe , O. Teschke, eds., Springer International Publishing, Cham, 2017, pp. 224--238, DOI 10.1007/978-3-319-62075-6_16 .
    Abstract
    Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open. In this paper we propose a solution -- to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexiformal framework, which has enough structure to support the various uses of models in scientific and technology workflows. Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroeck's drift-diffusion model for one-dimensional devices. This formalization -- and future extensions -- allows us to support the modeler by e.g. flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating MMS as "research data” and opens the way towards more MKM services for models.

  • B. Drees, Th. Koprucki, Make the most of your visual simulation data: The TIB AV-Portal enables citation of simulation results and increases their visibility, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 227-- 228, DOI 10.1109/NUSOD.2017.8010074 .
    Abstract
    There still exists no common standard on how to publish and cite visualisations of simulation data. We introduce the TIB AV-Portal as a sustainable infrastructure for audio-visual data using a combination of digital object identifiers (DOI) and media fragment identifiers (MFID) to cite these data in accordance with scientific standards. The benefits and opportunities of enhancing publications with visual data are illustrated by showing a use case from opto-electronics.

  • P. Farrell, Th. Koprucki, J. Fuhrmann, Comparison of consistent flux discretizations for drift diffusion beyond boltzmann statistics, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 219--220, DOI 10.1109/NUSOD.2017.8010070 .

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 397--405, DOI 10.1007/978-3-319-57394-6_42 .

  • M. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and electro-optical models for simulation of dynamics in broad-area semiconductor lasers, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 231--232.

  • S. Reichelt, Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012013/1--012013/15.
    Abstract
    We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms, and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1, other species may diffuse much slower, namely, with order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

  • M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Modeling and simulation of injection dynamics for quantum dot based single-photon sources, in: Proceedings of the 16th International Conference on Numerical Simulation of Optoelectronic Devices, J. Piprek, Ch. Poulton, M. Steel, M. DE Sterke, eds., IEEE Conference Publications Management Group, Piscataway, 2016, pp. 219--220.
    Abstract
    Single semiconductor quantum dots embedded in p-i-n diodes have been demonstrated to operate as electrically driven single-photon sources. By means of numerical simulations one can explore the limitations in the carrier injection dynamics and further improve the device technology. We propose a comprehensive modeling approach coupling the macroscopic transport of bulk carriers with an open quantum system to describe the essential physics of such devices on multiple scales.

  • M. Kantner, U. Bandelow, Th. Koprucki, J.-H. Schulze, A. Strittmatter, H.-J. Wünsche, On current injection into single quantum dots through oxide-confined PN-diodes, in: Proceedings of the 16th International Conference on Numerical Simulation of Optoelectronic Devices, J. Piprek, Ch. Poulton, M. Steel, M. DE Sterke, eds., IEEE Conference Publications Management Group, Piscataway, 2016, pp. 215--216.
    Abstract
    Current injection into single quantum dots embedded in vertical pn-diodes featuring oxide apertures is essential to the technological realization of single-photon sources. This requires efficient electrical pumping of sub-micron sized regions under pulsed excitation to achieve control of the carrier population of the desired quantum dots. We show experimentl and theoretical evidence for a rapid lateral spreading of the carriers after passing the ocide aperture in the conventional p-i-n-design in the low-injection regime suitable for single-photon emitters. By an alternative design employing p-doping up to the oxide aperture the current spreading can be suppressed resulting in an enhanced current confinement and increased injection efficiencies.

  • G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1--012009/20.
    Abstract
    This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.

  • M. Thomas, E. Bonetti, E. Rocca, R. Rossi, A rate-independent gradient system in damage coupled with plasticity via structured strains, in: Gradient Flows: From Theory to Application, B. Düring, C.-B. Schönlieb, M.-Th. Wolfram, eds., 54 of ESAIM Proceedings and Surveys, EDP Sciences, 2016, pp. 54--69.
    Abstract
    This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers.

  • P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann, Th. Koprucki, Numerical methods for drift-diffusion models, in: Handbook of Optoelectronic Device Modeling and Simulation (Two-Volume Set), J. Piprek, ed., Series in Optics and Optoelectronics, CRC Press Taylor & Francis Group, 2017.
    Abstract
    The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions.

  • TH. Koprucki, K. Tabelow, Mathematical models: A research data category?, in: Mathematical Software -- ICMS 2016: 5th International Conference, Berlin, Germany, July 11--14, 2016, Proceedings, G.-M. Greuel, Th. Koch, P. Paule, A. Sommese, eds., Lecture Notes in Computer Science, Springer International Publishing AG Switzerland, Cham, 2016, pp. 423--428.
    Abstract
    Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines and application areas. It is common to categorize the involved numerical data and to some extend the corresponding scientific software as research data. Both have their origin in mathematical models. In this contribution we propose a holistic approach to research data in MMS by including the mathematical models and discuss the initial requirements for a conceptual data model for this field.

  • A. Mielke, R. Rossi, G. Savaré, Balanced-Viscosity solutions for multi-rate systems, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26.
    Abstract
    Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients εα and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.

  • A. Mielke, Deriving effective models for multiscale systems via evolutionary $Gamma$-convergence, in: Control of Self-Organizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235--251.

  • A. Mielke, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. In Honor of Michael Ortiz's 60th Birthday, K. Weinberg, A. Pandolfi, eds., 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 135--160.
    Abstract
    In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.

    Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature.

    We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

  • A. Mielke, Relaxation of a rate-independent phase transformation model for the evolution of microstructure, in: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure, Workshop, March 14--18, 2016, R. Kienzler, D.L. Mcdowell, S. Müller, E.A. Werner, eds., 13 of Oberwolfach Reports, European Mathematical Society, 2016, pp. 840---842.

Preprints, Reports, Technical Reports

  • O. Burylko, A. Mielke, M. Wolfrum, S. Yanchuk, Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling, Preprint no. 2447, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2447 .
    Abstract, PDF (3022 kByte)
    We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i. e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.

  • TH. Koprucki, M. Kohlhase, K. Tabelow, D. Müller, F. Rabe, Model pathway diagrams for the representation of mathematical models, Preprint no. 2431, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2431 .
    Abstract, PDF (421 kByte)
    Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the van Roosbroeck system describing the carrier transport in semiconductors by drift and diffusion. We introduce an approach for the block-based composition of models from simpler components.

  • B. Drees, A. Kraft, Th. Koprucki, Reproducible research through persistently linked and visualized data, Preprint no. 2430, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2430 .
    Abstract, PDF (1945 kByte)
    The demand of reproducible results in the numerical simulation of opto-electronic devices or more general in mathematical modeling and simulation requires the (long-term) accessibility of data and software that were used to generate those results. Moreover, to present those results in a comprehensible manner data visualizations such as videos are useful. Persistent identifier can be used to ensure the permanent connection of these different digital objects thereby preserving all information in the right context. Here we give an overview over the state-of-the art of data preservation, data and software citation and illustrate the benefits and opportunities of enhancing publications with visual simulation data by showing a use case from opto-electronics.

  • 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, Preprint no. 2428, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2428 .
    Abstract, PDF (257 kByte)
    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, Optimal velocity control of a viscous Cahn--Hilliard system with convection and dynamic boundary conditions, Preprint no. 2427, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2427 .
    Abstract, PDF (306 kByte)
    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 an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fréchet differentiability of the associated control-to-state mapping in suitable Banach spaces, and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort.

  • P. Farrell, M. Patriarca, J. Fuhrmann, Th. Koprucki, Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics, Preprint no. 2424, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2424 .
    Abstract, PDF (1572 kByte)
    We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).

  • M. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices, Preprint no. 2421, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2421 .
    Abstract, PDF (246 kByte)
    We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

  • L. Donati, M. Heida, M. Weber, B. Keller, Estimation of the infinitesimal generator by square-root approximation, Preprint no. 2416, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2416 .
    Abstract, PDF (1739 kByte)
    For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.

  • P. Gurevich, S. Reichelt, Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients, Preprint no. 2413, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2413 .
    Abstract, PDF (1751 kByte)
    This paper is devoted to pulse solutions in FitzHugh-Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh-Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.

  • M. Kantner, M. Mittnenzweig, Th. Koprucki, Hybrid quantum-classical modeling of quantum dot devices, Preprint no. 2412, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2412 .
    Abstract, PDF (5602 kByte)
    The design of electrically driven quantum dot devices for quantum optical applications asks for modeling approaches combining classical device physics with quantum mechanics. We connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems to meet this requirement. By coupling the van Roosbroeck system with a quantum master equation in Lindblad form, we obtain a new hybrid quantum-classical modeling approach, which enables a comprehensive description of quantum dot devices on multiple scales: It allows the calculation of quantum optical figures of merit and the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way. We construct the interface between both theories in such a way, that the resulting hybrid system obeys the fundamental axioms of (non-)equilibrium thermodynamics. We show that our approach guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics. The feasibility of the approach is demonstrated by numerical simulations of an electrically driven single-photon source based on a single quantum dot in the stationary and transient operation regime.

  • M. Liero, S. Melchionna, The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems, Preprint no. 2411, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2411 .
    Abstract, PDF (392 kByte)
    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.

  • R. Rossi, M. Thomas, From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination, Preprint no. 2409, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2409 .
    Abstract, PDF (291 kByte)
    We revisit the weak, energetic-type existence results obtained in [Rossi/Thomas-ESAIM-COCV-21(1):1-59,2015] for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.

  • P. Dupuis, V. Laschos, K. Ramanan, Exit time risk-sensitive stochastic control problems related to systems of cooperative agents, Preprint no. 2407, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2407 .
    Abstract, PDF (443 kByte)
    We study sequences, parametrized by the number of agents, of exit time stochastic control problems with risk-sensitive costs structures generate by unbounded costs. We identify a fully characterizing assumption, under which, each of them corresponds to a risk-neutral stochastic control problem with additive cost, and also to a risk-neutral stochastic control problem on the simplex, where the specific information about the state of each agent can be discarded. We finally prove that, under some additional assumptions, the sequence of value functions converges to the value function of a deterministic control problem.

  • A. Mielke, Three examples concerning the interaction of dry friction and oscillations, Preprint no. 2405, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2405 .
    Abstract, PDF (3492 kByte)
    We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with state-dependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved.

  • M. Heida, On convergences of the squareroot approximation scheme to the Fokker--Planck operator, Preprint no. 2399, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2399 .
    Abstract, PDF (506 kByte)
    We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

  • M. Thomas, A comparison of delamination models: Modeling, properties, and applications, Preprint no. 2393, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2393 .
    Abstract, PDF (140 kByte)
    This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed.

  • P. Colli, G. Gilardi, J. Sprekels, On a Cahn--Hilliard system with convection and dynamic boundary conditions, Preprint no. 2391, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2391 .
    Abstract, PDF (291 kByte)
    This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn--Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn--Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo--Galerkin scheme, is introduced and rigorously discussed.

  • S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$-regularization, Preprint no. 2388, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2388 .
    Abstract, PDF (566 kByte)
    We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems.

  • P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with phase transition, Preprint no. 2384, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2384 .
    Abstract, PDF (296 kByte)
    In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.

  • A.F.M. TER Elst, J. Rehberg, Consistent operator semigroups and their interpolation, Preprint no. 2382, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2382 .
    Abstract, PDF (231 kByte)
    Under a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Preprint no. 2371, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2371 .
    Abstract, PDF (598 kByte)
    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

  • P. Colli, J. Sprekels, Optimal boundary control of a nonstandard Cahn--Hilliard system with dynamic boundary condition and double obstacle inclusions, Preprint no. 2370, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2370 .
    Abstract, PDF (268 kByte)
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P.Podio-Guidugli in Ric. Mat. 55 (2006), pp.105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1-30, for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

  • P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions, Preprint no. 2369, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2369 .
    Abstract, PDF (353 kByte)
    This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

  • M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Preprint no. 2366, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2366 .
    Abstract, PDF (548 kByte)
    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.

  • P. Gussmann, A. Mielke, Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks, Preprint no. 2359, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2359 .
    Abstract, PDF (428 kByte)
    The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Gamma converge to the linearized elastic energy with a local constraint of noninterpenetrability along the crack.

  • H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems, Preprint no. 2356, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2356 .
    Abstract, PDF (454 kByte)
    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.

  • M. Heida, R.I.A. Patterson, D.R.M. Renger, The space of bounded variation with infinite-dimensional codomain, Preprint no. 2353, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2353 .
    Abstract, PDF (600 kByte)
    We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.

  • A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities, Preprint no. 2349, WIAS, Berlin, 2016.
    Abstract, PDF (618 kByte)
    We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

  • S. Bommer, R. Seemann, S. Jachalski, D. Peschka, B. Wagner, Liquid-liquid dewetting: Morphologies and rates, Preprint no. 2346, WIAS, Berlin, 2016.
    Abstract, PDF (1107 kByte)
    The dependence of the dissipation on the local details of the flow field of a liquid polymer film dewetting from a liquid polymer substrate is shown, solving the free boundary problem for a two-layer liquid system. As a key result we show that the dewetting rates of such a liquid bi-layer system can not be described by a single power law but shows transient behaviour of the rates, changing from increasing to decreasing behaviour. The theoretical predictions on the evolution of morphology and rates of the free surfaces and free interfaces are compared to measurements of the evolution of the polystyrene(PS)-air, the polymethyl methacrylate (PMMA)-air and the PS-PMMA interfaces using in situ atomic force microscopy (AFM), and they show excellent agreement.

  • A. Mielke, Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, Preprint no. 2326, WIAS, Berlin, 2016.
    Abstract, PDF (444 kByte)
    We consider reaction-diffusion systems on a bounded domain with no-flux boundary conditions. All reactions are given by the mass-action law and are assumed to satisfy the complex-balance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing.
     
    We discuss three methods to obtain energy-dissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the log-Sobolev estimate and suitable handling of the reaction terms as well as the mass-conservation relations. The three methods are (i) a convexification argument based on the author's joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument of developed by Glitzky and Hünlich.

  • L. Heltai, N. Rotundo, Error estimates in weighted Sobolev norms for finite element immersed interface methods, Preprint no. 2323, WIAS, Berlin, 2016.
    Abstract, PDF (1105 kByte)
    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 a uniform 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.

  • P.-É. Druet, Analysis of improved Nernst--Planck--Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix, Preprint no. 2321, WIAS, Berlin, 2016.
    Abstract, PDF (387 kByte)
    We continue our investigations of the improved Nernst-Planck-Poisson model introduced by Dreyer, Guhlke and Müller 2013. In the paper by Dreyer, Druet, Gajewski and Guhlke 2016, the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for the mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of emphdifferences of chemical potentials that supports the modelling and the analysis in Dreyer, Druet, Gajewski and Guhlke 2016. We prove the global-in-time existence in this solution class.

  • A. Mielke, D. Peschka, N. Rotundo, M. Thomas, Gradient structure for optoelectronic models of semiconductors, Preprint no. 2317, WIAS, Berlin, 2016.
    Abstract, PDF (178 kByte)
    We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon- densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately

  • K. Disser, Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction, Preprint no. 2313, WIAS, Berlin, 2016.
    Abstract, PDF (302 kByte)
    We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic” diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces.

  • D. Horstmann, J. Rehberg, H. Meinlschmidt, The full Keller--Segel model is well-posed on fairly general domains, Preprint no. 2312, WIAS, Berlin, 2016.
    Abstract, PDF (369 kByte)
    In this paper we prove the well-posedness of the full Keller-Segel system, a quasilinear strongly coupled reaction-crossdiffusion system, in the spirit that it always admits a unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. Apparently, there exists no comparable existence result for the full Keller-Segel system up to now. The proof is carried out for general source terms and is based on recent nontrivial elliptic and parabolic regularity results which hold true even on fairly general spatial domains, combined with an abstract solution theorem for nonlocal quasilinear equations by Amann.

  • A. Muntean, S. Reichelt, Corrector estimates for a thermo-diffusion model with weak thermal coupling, Preprint no. 2310, WIAS, Berlin, 2016.
    Abstract, PDF (1060 kByte)
    The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the “high-contrast” is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes

  • 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, Preprint no. 2309, WIAS, Berlin, 2016.
    Abstract, PDF (374 kByte)
    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.

  • P. Colli, G. Gilardi, J. Sprekels, Optimal boundary control of a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition, Preprint no. 2307, WIAS, Berlin, 2016.
    Abstract, PDF (347 kByte)
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fr&aecute;chenett differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.

  • M. Kantner, Th. Koprucki, Numerical simulation of carrier transport in semiconductor devices at cryogenic temperatures, Preprint no. 2296, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2296 .
    Abstract, PDF (1445 kByte)
    At cryogenic temperatures the electron-hole plasma in semiconductor materials becomes strongly degenerate, leading to very sharp internal layers, extreme depletion in intrinsic domains and strong nonlinear diffusion. As a result, the numerical simulation of the drift-diffusion system suffers from serious convergence issues using standard methods. We consider a one-dimensional p-i-n diode to illustrate these problems and present a simple temperature-embedding scheme to enable the numerical simulation at cryogenic temperatures. The method is suitable for forward-biased devices as they appear e.g. in optoelectronic applications.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible reacting electrolytes, Preprint no. 2291, WIAS, Berlin, 2016.
    Abstract, PDF (638 kByte)

    We consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce 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 the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in- time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.
  • K. Disser, A.F.M. TER Elst, J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, Preprint no. 2249, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2249 .
    Abstract, PDF (335 kByte)
    We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

  • H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions, Preprint no. 2238, WIAS, Berlin, 2016.
    Abstract, PDF (14 MByte)
    This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

  • J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, Preprint no. 2233, WIAS, Berlin, 2016.
    Abstract, PDF (436 kByte)
    We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L1 using Cziszar-Kullback-Pinsker type inequalities.

  • K. Disser, M. Liero, J. Zinsl, On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions, Preprint no. 2227, WIAS, Berlin, 2016.
    Abstract, PDF (489 kByte)
    We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

  • M. Erbar, M. Fathi, V. Laschos, A. Schlichting, Gradient flow structure for McKean--Vlasov equations on discrete spaces, Preprint no. 2219, WIAS, Berlin, 2016.
    Abstract, PDF (393 kByte)
    In this work, we show that a family of non-linear mean-field equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity

  • M. Liero, A. Mielke, G. Savaré, Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures, Preprint no. 2207, WIAS, Berlin, 2016.
    Abstract, PDF (1154 kByte)
    We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

  • H. Gajewski, M. Liero, R. Nürnberg, H. Stephan, WIAS-TeSCA -- Two-dimensional semi-conductor analysis package, Technical Report no. 14, WIAS, Berlin, 2016.
    Abstract
    WIAS-TeSCA (Two-dimensional semiconductor analysis package) is a simulation tool for the numerical simulation of charge transfer processes in semiconductor structures, especially in semiconductor lasers. It is based on the drift-diffusion model and considers a multitude of additional physical effects, like optical radiation, temperature influences and the kinetics of deep impurities. Its efficiency is based on the analytic study of the strongly nonlinear system of partial differential equations -- the van Roosbroeck system -- which describes the electron and hole currents. Very efficient numerical procedures for both the stationary and transient simulation have been implemented.
    WIAS-TeSCA has been successfully used in the research and industrial development of new electronic and optoelectronic semiconductor devices such as transistors, diodes, sensors, detectors and lasers and has already proved its worth many times in the planning and optimization of these devices. It covers a broad spectrum of applications, from hetero-bipolar transistor (mobile telephone systems, computer networks) through high-voltage transistors (power electronics) and semiconductor laser diodes (fiber optic communication systems, medical technology) to radiation detectors (space research, high energy physics).
    WIAS-TeSCA is an efficient simulation tool for analyzing and designing modern semiconductor devices with a broad range of performance that has proved successful in solving many practical problems. Particularly, it offers the possibility to calculate self-consistently the interplay of electronic, optical and thermic effects.

Talks, Poster

  • S. Reichelt, Corrector estimates for elliptic and parabolic equations with periodic coefficients, Analysis Seminar, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Angewandte Mathematik, Erlangen, May 18, 2017.

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

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

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

  • S. Reichelt, Traveling waves in FitzHugh--Nagumo systems with rapidly oscillating coefficients, Workshop ``Control of Self-organizing Nonlinear Systems'', August 29 - 31, 2017, Collaborative Research Center 910: Control of self-organizing nonlinear systems: Theoretical methods and concepts of application, Lutherstadt Wittenberg, August 30, 2017.

  • M. Heida, A. Mielke, Effective models for interfaces with many scales, CRC 1114 Conference ''Scaling Cascades in Complex Systems 2017'', Berlin, March 27 - 29, 2017.

  • M. Heida, Averaging of time-periodic dissipation potentials in rate-independent processes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 9, 2017.

  • M. Heida, On G-convergence and stochastic two-scale convergences of the squareroot approximation scheme to the Fokker--Planck operator, GAMM-Workshop on Analysis of Partial Differential Equations, September 27 - 29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

  • M. Heida, What is ... GENERIC?, CRC1114 Colloquium, Freie Universität Berlin, SFB 1114, June 29, 2017.

  • M. Liero, Modeling and simulation of electrothermal feedback in large-area organic LEDs, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Light-Emitting Diodes'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 25, 2017.

  • M. Liero, On entropy-transport problems and the Hellinger--Kantorovich distance, Seminar of Team EDP-AIRSEA-CVGI, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble, France, January 26, 2017.

  • M. Mittnenzweig, An entropic gradient structure Lindblad equations, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 9, 2017.

  • M. Mittnenzweig, An entropic gradient structure for quantum Markov semigroups, Workshop ``Applications of Optimal Transportation in the Natural Sciences'', January 30 - February 3, 2017, Mathematisches Forschungszentrum Oberwolfach, January 31, 2017.

  • CH. Mukherjee, Asymptotic behavior of the mean-field polaron, Probability and Mathematical Physics Seminar, Courant Institute of Mathematical Sciences, Department of Mathematics, New York, USA, March 20, 2017.

  • D. Peschka, Doping optimization for optoelectronic devices, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), Post-Deadline session, July 27 - 28, 2017, Technical University, Lyngby Campus, Kopenhagen, Denmark, July 28, 2017.

  • D. Peschka, Modelling and simulation of suspension flow, Graduate Seminar PDE in the Sciences, Universität Bonn, Institut für Angewandte Mathematik, January 20, 2017.

  • D. Peschka, Motion of thin droplets over surfaces, Making a Splash -- Driplets, Jets and Other Singularities, March 20 - 24, 2017, Brown University, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, USA, March 22, 2017.

  • D. Peschka, Variational structure of fluid motion with contact lines in thin-film models, Kolloquium Angewandte Mathematik, Universität der Bundeswehr, München, May 31, 2017.

  • A. Fischer, M. Liero, A. Glitzky, Th. Koprucki, K. Vandewal, S. Lenk, S. Reinicke, Predicting electrothermal behavior from lab-size OLEDs to large area lighting panels, MRS Spring Meeting, Phoenix, Arizona, USA, April 17 - 21, 2017.

  • A. Glitzky, Electrothermal description of organic semiconductor devices by $p(x)$-Laplace thermistor models, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 9, 2017.

  • M. Thomas, Dynamics of rock dehydration on multiple scales, SCCS Days, November 8 - 10, 2017, SFB 1114 ``Scaling Cascades in Complex Systems'', Schmöckwitz, November 9, 2017.

  • M. Thomas, Mathematical modeling and analysis of evolution processes in solids and the influence of bulk-interface-interaction, Humboldt-Universität zu Berlin, Institut für Mathematik, October 20, 2017.

  • M. Thomas, Rate-independent delamination processes in visco-elasticity, Miniworkshop on Dislocations, Plasticity, and Fracture, February 13 - 16, 2017, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, February 15, 2017.

  • M. Thomas, Why scientist in Academia?, I, SCIENTIST: The Conference on Gender, Career Paths and Networking, May 12 - 14, 2017, Freie Universität, Berlin, May 14, 2017.

  • M. Thomas, tba, ECCOMAS Thematic Conference ``Modern Finite Element Technologies 2017'', August 21 - 25, 2017, Schwerpunktprogramm SPP 1748, Bad Honnef.

  • M. Thomas, tba, Variational Methods for Evolution, November 12 - 17, 2017, Mathematisches Forschungszentrum Oberwolfach, November 13, 2017.

  • M. Thomas, tba, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, Centro de Matemática, Lisboa, Portugal.

  • D.H. Doan, A unified Scharfetter Gummel scheme, Halbleiterseminar, WIAS, March 2, 2017.

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

  • P.-É. Druet, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible electrolytes, Seminar EDE, Institute of Mathematics, Department of Evolution Differential Equations (EDE), Prague, Czech Republic, January 10, 2017.

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-laplace thermistor models, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), Villeneuve d'Ascq, France, June 15, 2017.

  • TH. Koprucki, Comparison of consistent flux discretizations for drift diffusion beyond Boltzmann statistics, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Numerical Methods'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 27, 2017.

  • TH. Koprucki, Halbleiter-Bauteilsimulation: Modelle und numerische Verfahren, Block-Seminar des SFB 787 ``Nanophotonik'', June 7 - 9, 2017, Technische Universität Berlin, Graal-Müritz, June 8, 2017.

  • TH. Koprucki, How to tidy up the jungle of mathematical models? A prerequisite for sustainable research software, 2nd Conference 2017 on Non-Textual Information ``Software and Services for Science (S3)'', May 10 - 11, 2017, Technische Informationsbibliothek, Hannover, May 11, 2017, DOI 10.5446/31023 .

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

  • TH. Koprucki, Mathematical models as research data in numerical simulation of opto-electronic devices, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Model Representation'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 27, 2017.

  • TH. Koprucki, Mathematical models as research data via flexiformal theory graphs, 10th Conference on Intelligent Computer Mathematics (CICM 2017), track ``Mathematical Knowledge Management'', July 16 - 21, 2017, University of Edinburgh, UK, July 20, 2017.

  • TH. Koprucki, Model Pathway Diagrams for knowledge representation in mathematical modeling and simulation, 19th ÖMG Congress and Annual DMV Meeting, minisymposium M3 ``From Information to Knowledge Management'', September 11 - 15, 2017, Paris-Lodron University of Salzburg, Austria, September 12, 2017.

  • TH. Koprucki, On current injection into single quantum dots through oxide-confined pn-diodes, 10th Annual Meeting ``Photonic Devices'', February 9 - 10, 2017, Zuse Institute Berlin (ZIB), Berlin, February 9, 2017.

  • TH. Koprucki, On the Scharfetter-Gummel scheme for the discretization of drift-diffusion equations and its generalization beyond Boltzmann, Kolloquium Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, Insittut für Mathematik, May 30, 2017.

  • TH. Koprucki, Über das L-Konzept einer physikalischen Theorie, Seminar Wissensrepräsentation und -verarbeitung, Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Informatik, Erlangen, May 17, 2017.

  • M. Liero, A. Glitzky, Th. Koprucki, J. Fuhrmann, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Multiscale Modelling of Organic Semiconductors: From Elementary Processes to Devices, Grenoble, France, September 12 - 15, 2017.

  • M. Liero, tba, Variational Methods for Evolution, November 12 - 17, 2017, Mathematisches Forschungszentrum Oberwolfach.

  • A. Mielke, A geometric approach to reaction-diffusion equations, Institutskolloquium, Universität Potsdam, Institut für Mathematik, January 25, 2017.

  • A. Mielke, Entropy-induced geometry for classical and quantum Markov semigroups, SMS Colloquium, University College Cork, School of Mathematical Science, Ireland, September 11, 2017.

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

  • A. Mielke, Interfaces with many scales, Second CRC 1114 Days ''Scaling Cascades in Complex Systems'', November 8 - 10, 2017, Freie Universität Berlin, Schmöckwitz, November 10, 2017.

  • A. Mielke, Mathematical modeling of semiconductors: From quantum mechanics to devices, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, Centro de Matemática, Lisboa, Portugal.

  • A. Mielke, On self-induced oscillations for friction reduction with applications to walking, Conference ``Dynamical Systems and Geometric Mechanics'', June 12 - 14, 2017, Technische Universität München, Zentrum für Mathematik, June 13, 2017.

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

  • A. Mielke, Oscillations in systems with hysteresis, SFB 910 Symposium ``Stability Versus Oscillations in Complex Systems'', Technische Universität Berlin, Institut für Theoretische Physik, February 10, 2017.

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

  • A. Mielke, Uniform exponential decay for energy-reaction-diffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italy, March 21, 2017.

  • M. Mittnenzweig, A variational approach to the Lindblad equations, Scientific computing seminar, Université École des Ponts, CERMICS, Paris, France, April 24, 2017.

  • M. Mittnenzweig, Gradient flow structures for quantum master equations, Analysis-Seminar Augsburg-München, Universität Augsburg, Institut für Mathematik, June 8, 2017.

  • M. Mittnenzweig, Variational methods for quantum master equations, Barcelona Graduate School of Mathematics Junior Meeting, October 9 - 10, 2017, Berlin Mathematical School, Spain, October 10, 2017.

  • M. Mittnenzweig, tba, Variational Methods for Evolution, November 12 - 17, 2017, Mathematisches Forschungszentrum Oberwolfach.

  • J. Rehberg, On optimal elliptic Sobolev regularity, Oberseminar Prof. Ira Neitzel, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, February 2, 2017.

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

  • K. Disser, Das Leibniz-Mentoring im Rückblick (Podiumsdiskussion), Leibniz-Mentoring Festakt, Leibniz-Mentoringprogramm für Wissenschaftlerinnen, Berlin, October 6, 2016.

  • K. Disser, E-convergence to the quasi-steady-state approximation in systems of chemical reactions, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

  • K. Disser, Zhukovskys Theorem in der Fluid-Struktur-Interaktion, Universität Augsburg, Institut für Mathematik, July 4, 2016.

  • S. Reichelt, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

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

  • S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

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

  • S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25 - 29, 2016.

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

  • S. Reichelt, On periodic homogenization, 20. Harz-Seminar zur Strukturbildung in Chemie und Biophysik, February 21 - 22, 2016, Physikalisch-Technische Bundesanstalt, Hahnenklee, February 22, 2016.

  • N. Rotundo, Numerical methods for drift-diffusion models, Seminar ``Angewandte Mathematik'', Ernst Moritz Arndt Universität Greifswald, Institut für Mathematik und Informatik, June 28, 2016.

  • N. Rotundo, On some extension of energy-drift-diffusion models, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 34 ``Mathematical Modeling of Charge Transport in Graphene and Low Dimensional Structure'', June 13 - 18, 2016, Universidade de Santiago de Compostela, Spain, June 14, 2016.

  • N. Rotundo, Thermodynamic modeling of optoelectronic semiconductor devices, Mathematical Models for Quantum and Classical Mechanics (SEMODAY2016), November 17 - 18, 2016, Università degli Studi di Firenze, Dipartamento di Matematica, Florence, Italy, November 18, 2016.

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

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

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

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

  • M. Heida, A. Mielke, Ch. Kraus, M. Thomas, Effective models for interfaces with many scales, SCCS Days, CRC 1114 ``Complex Processes involving Cascades of Scales'', Ketzin, October 10 - 12, 2016.

  • M. Heida, Homogenization of the random conductance model, 7th European Congress of Mathematics (ECM), session ``Probability, Statistics and Financial Mathematics'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

  • M. Heida, Homogenization of the random conductance model, Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 26 - 28, 2016, Technische Universität Dortmund, Fachbereich Mathematik, Dortmund, September 26, 2016.

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

  • M. Heida, On homogenization of rate-independent systems, sc Matheon Multiscale Seminar, Technische Universität Berlin, February 17, 2016.

  • M. Heida, Stochastic homogenization of 1-homogeneous functionals, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • M. Heida, Stochastic homogenization of rate-independent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

  • M. Heida, Stochastic homogenization of rate-independent systems, Joint Annual Meeting of DMV and GAMM, Session ``Multiscales and Homogenization'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

  • M. Liero, Electrothermal modeling of large-area OLEDs, sc Matheon Center Days, April 11 - 12, 2016, Technische Universität Berlin, Institut für Mathematik, Berlin, April 11, 2016.

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

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

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

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

  • M. Liero, On Entropy-Transport problems and distances between positive measures, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

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

  • M. Liero, On entropy-transport problems and the Hellinger--Kantorovich distance, Follow-up Workshop to Junior Hausdorff Trimester Program ``Optimal Transportation'', August 29 - September 2, 2016, Hausdorff Research Institute for Mathematics, Bonn, August 30, 2016.

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

  • M. Mittnenzweig, Gradient structures for Lindblad equations satisfying detailed balance, 3rd PhD Workshop, May 30 - 31, 2016, International Research Training Group of the Collaborative Research Center (SFB) 1114 ``Scaling Cascades in Complex Systems'', Güstrow, May 31, 2016.

  • M. Mittnenzweig, Zwischen Flüssigkristallen und Informationstheorie, Kurs ``Eine naturwissenschaftliche Reise hinter das Display eines Smartphones'', July 4 - 8, 2016, Deutsche Schülerakademie, Braunschweig.

  • H. Neidhardt, To the spectral theory of vector-valued Sturm--Liouville operators with summable potentials and point interactions, Workshop ``Mathematical Challenge of Quantum Transport in Nanosystems'' (Pierre Duclos Workshop), November 14 - 15, 2016, Saint Petersburg National Research University of Informational Technologies, Mechanics, and Optics, Russian Federation, November 15, 2016.

  • D. Peschka, A free boundary problem for the flow of viscous liquid bilayers, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • D. Peschka, A free boundary problem for the motion of viscous liquids, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • D. Peschka, Mathematical modeling, analysis, and optimization of strained germanium-microbridges, sc Matheon Center Days, April 11 - 12, 2016, Technische Universität Berlin, Institut für Mathematik, Berlin, April 12, 2016.

  • D. Peschka, Multi-phase flows with contact lines: Solid vs liquid substrates, Industrial and Applied Mathematics Seminar, University of Oxford, Mathematical Institute, UK, October 27, 2016.

  • D. Peschka, Scientific computing and applied mathematics for optoelectronics and soft matter problems, Universiteit Twente, Departement of Mathematics, Enschede, Netherlands, November 8, 2016.

  • D. Peschka, Thin film free boundary problems --- Modeling of contact line dynamics with gradient formulations, CeNoS-Kolloquium, Westfälische Wilhelms-Universität Münster, Center for Nonlinear Science, January 12, 2016.

  • D. Peschka, Towards the optimization of Ge micro bridges, The 19th European Conference on Mathematics for Industry (ECMI 2016), minisymposium ``Charge Transport in Semiconductor Materials: Emerging and Established Mathematical Topics'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Faculty of Mathematics, Santiago de Compostela, Spain, June 15, 2016.

  • D. Peschka, Towards the optimization of on-chip germanium lasers, sc Matheon Workshop, 9th Annual Meeting ``Photonic Devices'', March 3 - 4, 2016, Zuse Institute Berlin, Berlin, March 4, 2016.

  • M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, M. Bär, A. Mielke, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • J. Sprekels, A new identification problem: Optimizing the order in a nonlocal evolution equation, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 13, 2016.

  • J. Sprekels, On a nonstandard Cahn--Hilliard system with dynamic boundary condition, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 9, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 9, 2016.

  • J. Sprekels, On a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions, Mathematical Institute of the Czech Academy of Sciences, Prague, December 6, 2016.

  • J. Sprekels, On a nonstandard viscous Cahn--Hilliard system: Existence and optimal control, Congress of the Italian Society of Industrial and Applied Mathematics (SIMAI 2016), September 13 - 16, 2016, Politecnico di Milano, Italy, September 14, 2016.

  • J. Sprekels, On a nonstandard viscous nonlocal Cahn--Hilliard system, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 23, 2016.

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

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

  • M. Thomas, Analysis and optimization for edge-emitting semiconductor heterostructures, 7th European Congress of Mathematics (ECM), session CS-8-A, July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 19, 2016.

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

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Delamination models in visco-elastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

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

  • M. Thomas, Energetic concepts for coupled rate-independent and rate-dependent processes: Damage & delamination in visco-elastodynamics, International Conference ``Mathematical Analysis of Continuum Mechanics and Industrial Applications II'' (CoMFoS16), October 22 - 24, 2016, Kyushu University, Fukuoka, Japan.

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

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

  • M. Thomas, Mathematische Modellierung, Analysis und Optimierung von Germanium-Lasern, Vortrag vor dem WGL-Präsidenten anlässlich seines WIAS-Besuchs, WIAS Berlin, Berlin, February 18, 2016.

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

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

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

  • D.H. Doan, Numerical methods in non-Boltzmann regimes, sc Matheon Workshop, 9th Annual Meeting ``Photonic Devices'', March 3 - 4, 2016, Zuse Institute Berlin, Berlin, March 4, 2016.

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

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

  • S.P. Frigeri, On some nonlocal diffuse-interface models for binary fluids: Regularity results and applications, Congress of the Italian Society of Industrial and Applied Mathematics (SIMAI 2016), September 13 - 16, 2016, Politecnico di Milano, Italy, September 14, 2016.

  • S.P. Frigeri, Optimal distributed control for nonlocal Cahn--Hilliard/Navier--Stokes systems in 2D, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20 - 24, 2016, Cortona, Italy, June 24, 2016.

  • S.P. Frigeri, Recent results on some diffuse-interface models for incompressible binary fluids with nonlocal interaction, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 23, 2016.

  • TH. Koprucki, How the WIAS handles mathematical research data, 1st Leibniz MMS Days, January 27 - 29, 2016, WIAS, January 28, 2016.

  • TH. Koprucki, Mathematical research data: How the WIAS handles mathematical research data, Joint Annual Meeting of DMV and GAMM, Zentralblatt Meeting on Mathematical Knowledge Management ``Mathematics on the WEB -- Information and Communication in Mathematics'', March 7 - 11, 2016, Technische Universität Braunschweig, March 9, 2016.

  • TH. Koprucki, On current injection into single quantum dots through oxide-confined PN-diodes, 16th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2016), July 7 - 17, 2016, University of Sydney, Sydney, Australia, July 14, 2016.

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

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

  • A. Mielke, Evolutionary Gamma-convergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29 - September 2, 2016, Humboldt-Universität zu Berlin, Institut für Mathematik.

  • A. Mielke, Evolutionary relaxation for a rate-independent phase-transformation model, Workshop ``Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure'', March 14 - 18, 2016, Mathematisches Forschungszentrum Oberwolfach, March 14, 2016.

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

  • A. Mielke, Global existence for finite-strain viscoplasticity via the energy-dissipation principle, Seminar ``Analysis & Mathematical Physics'', Institute of Science and Technology Austria (IST Austria), Vienna, Austria, July 7, 2016.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion equation, Mathematisches Kolloquium, Westfälische Wilhelms-Universität, Institut für Mathematik, Münster, April 28, 2016.

  • A. Mielke, Microstructure evolution via relaxation for a rate-independent elastic two-phase model, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

  • A. Mielke, Multiscale modeling via evolutionary Gamma convergence, 3rd PhD Workshop, May 30 - 31, 2016, International Research Training Group of the Collaborative Research Center (SFB) 1114 ``Scaling Cascades in Complex Systems'', Güstrow, May 30, 2016.

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

  • A. Mielke, On a model for the evolution of microstructures in solids -- Evolutionary relaxation, KTGU-IMU Mathematics Colloquia, March 30 - 31, 2016, Kyoto University, Department of Mathematics, Japan, March 31, 2016.

  • A. Mielke, On entropic gradient structures for classical and quantum Markov processes with detailed balance, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 11, 2016.

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

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

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

  • A. Mielke, Rate-independent microstructure evolution via relaxation of a two-phase model, Workshop ``Advances in the Mathematical Analysis of Material Defects in Elastic Solids'', June 6 - 10, 2016, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, June 10, 2016.

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

  • J. Rehberg, On non-smooth parabolic equations, Oberseminar Analysis, Universität Kassel, Institut für Mathematik, May 2, 2016.

  • J. Rehberg, On non-smooth parabolic equations, Oberseminar Analysis, Leibniz Universität Hannover, Institut für Angewandte Mathematik, May 10, 2016.

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

External Preprints

  • J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, Preprint no. arXiv:1511.02376v2, Cornell University Library, arXiv.org, 2016.
    Abstract
    A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.