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search.filters.applied.f.access: openAccess × Author: Rehberg, Joachim × Start date: 2000 × DDC: 510 × Has files: Yes × Type: Text × Version: publishedVersion ×

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Now showing 1 - 10 of 38
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    Maximal parabolic regularity for divergence operators on distribution spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, Joachim
    We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented
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    Hölder-estimates for non-autonomous parabolic problems with rough data
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Meinlschmidt, Hannes; Rehberg, Joachim
    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.
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    Hardys 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, Joachim
    We develop a geometric framework for Hardys inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.
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    A 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, Joachim
    An 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.
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    Coercivity for elliptic operators and positivity of solutions on Lipschitz domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, Joachim
    We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of 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
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    On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Chill, Ralph; Meinlschmidt, Hannes; Rehberg, Joachim
    We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.
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    L ∞-estimates for divergence operators on bad domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Elst, A.F.M. ter; Rehberg, Joachim
    In this paper, we prove L^infty-estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.
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    Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Meinlschmidt, Hannes; Rehberg, Joachim
    In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order 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.
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    Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Kaiser, Hans-Christoph; Rehberg, Joachim
    We investigate optimal elliptic regularity of anisotropic div-grad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces.
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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