Browsing by Author "Rehberg, Joachim"
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- ItemThe 3D transient semiconductor equations with gradient-dependent and interfacial recombination(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Disser, Karoline; Rehberg, JoachimWe establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.
- ItemAnalyticity for some operator functions from statistical quantum mechanics : dedicated to Günter Albinus(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hoke, Kurt; Kaiser, Hans-Christoph; Rehberg, Joachim; Albinus, GünterFor rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.
- ItemBlow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, MichaelLiteraturverz.
- ItemBoundary coefficient control : a maximal parabolic regularity approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Hömberg, Dietmar; Krumbiegel, Klaus; Rehberg, JoachimWe investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
- ItemClassical solutions of drift-diffusion equations for semiconductor devices: the 2D case(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Kaiser, Hans-Christian; Neidhardt, Hagen; Rehberg, Joachim; Gajewski, Herbert; Gröger, Konrad; Zacharias, KlausWe regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.
- ItemCoercivity for elliptic operators and positivity of solutions on Lipschitz domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, JoachimWe show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures
- ItemConsistent operator semigroups and their interpolation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Elst, A.F.M. ter; Rehberg, JoachimUnder a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators.
- ItemA criterion for a two-dimensional domain to be Lipschitzian(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Rehberg, JoachimWe prove that a two-dimensional domain is already Lipschitzian if only its boundary admits locally a one-dimensional, bi-Lipschitzian parametrization.
- ItemDirect computation of elliptic singularities across anisotropic, multi-material edges(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Kaiser, Hans-Christoph; Rehberg, JoachimWe characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and fourmaterial edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark Lshape problem.
- ItemEssential boundedness for solutions of the Neumann problem on general domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Meinlschmidt, Hannes; Rehberg, JoachimLet the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.
- ItemExtrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Meinlschmidt, Hannes; Rehberg, JoachimIn this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.
- ItemThe full Keller-Segel model is well-posed on fairly general domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Horstmann, Dirk; Rehberg, Joachim; Meinlschmidt, HannesIn 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. Nous considèrons le système de Keller et Segel dans son intégralité, un système quasilinéaire à réaction-diffusion fortement couplé. Le résultat principal montre que ce syst`eme est bien posé, cest-à-dire il admet une solution unique existant localement en temps à valeurs dans un espace fonctionnel approprié, pourvu que les valeurs initiales sont réguliers. Apparemment, il nexiste pas encore des résultats comparables. Pour la demonstration, nous utilisons des résultats récents de régularité elliptique et parabolique applicable à des domaines assez générals, combiné avec un théorème abstrait dAmann concernant les équations quasilinéaires non locales.
- ItemHardys inequality for functions vanishing on a part of the boundary(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Egert, Moritz; Haller-Dintelmann, Robert; Rehberg, JoachimWe develop a geometric framework for Hardys inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.
- ItemHölder continuity for second order elliptic problems with nonsmooth data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Haller-Dintelmann, Robert; Meyer, Christian; Rehberg, JoachimThe well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.
- ItemHölder estimates for parabolic operators on domains with rough boundary(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Disser, Karoline; Rehberg, Joachim; Elst, A.F.M. terIn 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.
- ItemHölder estimates for second-order operators with mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) ter Elst, A.F.M.; Rehberg, JoachimIn this paper we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain
- ItemHölder-estimates for non-autonomous parabolic problems with rough data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Meinlschmidt, Hannes; Rehberg, JoachimIn 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.
- ItemA Kohn-Sham system at zero temperature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Cornean, Horia; Hoke, Kurt; Neidhardt, Hagen; Racec, Paul Nicolae; Rehberg, JoachimAn one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
- ItemL ∞-estimates for divergence operators on bad domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Elst, A.F.M. ter; Rehberg, JoachimIn this paper, we prove L^infty-estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.
- ItemMaximal parabolic regularity for divergence operators including mixed boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Haller-Dintelmann, Robert; Rehberg, JoachimWe show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.