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Author: Rehberg, Joachim × End date: 2019 × Start date: 2019 × End date: 2018 × Start date: 2010 × DDC: 510 × Type: Text × Version: publishedVersion ×

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Now showing 1 - 10 of 19
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    Optimal elliptic Sobolev regularity near three-dimensional, multi-material Neumann vertices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Haller-Dintelmann, Robert; Höppner, Wolfgang; Kaiser, Hans-Christoph; Rehberg, Joachim; Ziegler, Günter M.
    We study relative stability properties of different clusters of closely packed one- and two-dimensional localized peaks of the Swift-Hohenberg equation. We demonstrate the existence of a 'spatial Maxwell' point where clusters are almost equally stable, irrespective of the number of pes involved. Above (below) the Maxwell point, clusters become more (less) stable with the increase of the number of peaks
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    A criterion for a two-dimensional domain to be Lipschitzian
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Rehberg, Joachim
    We prove that a two-dimensional domain is already Lipschitzian if only its boundary admits locally a one-dimensional, bi-Lipschitzian parametrization.
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    Blow-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, Michael
    Literaturverz.
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    Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Disser, Karoline; Kaiser, Hans-Christoph; Rehberg, Joachim
    On bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail.
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    The 3D transient semiconductor equations with gradient-dependent and interfacial recombination
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Disser, Karoline; Rehberg, Joachim
    We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on 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.
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    Parabolic equations with dynamical boundary conditions and source terms on interfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elst, A.F.M. ter; Meyries, Martin; Rehberg, Joachim
    We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable Lp-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.
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    Hölder estimates for second-order operators with mixed boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) ter Elst, A.F.M.; Rehberg, Joachim
    In 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
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    Hölder 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. ter
    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.
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    The 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, Hannes
    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. 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.
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    On maximal parabolic regularity for non-autonomous parabolic operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Disser, Karoline; Elst, A.F.M. ter; Rehberg, Joachim
    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.