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- ItemSobolev-Morrey spaces associated with evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.In this text we introduce new classes of Sobolev-Morrey spaces being adequate for the regularity theory of second order parabolic boundary value problems on Lipschitz domains of space dimension n ≥ 3 with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned Sobolev-Morrey spaces of solutions and right hand sides.
- ItemMaximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.This text is devoted to maximal regularity results for second order parabolic systems on Lipschitz domains of space dimension n ≥ 3 with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of Sobolev–Morrey spaces for solutions and right hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.
- ItemAnalytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bugert, Beatrice; Schmidt, GuntherThis paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a 2π-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
- ItemAn integral equation approach for electromagnetic scattering by biperiodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bugert, BeatriceThe objective of this paper is the analytical investigation of an integral equation formulation for electromagnetic scattering by 2π-biperiodic multilayered structures with polyhedral Lipschitz regular interfaces. Extending the combined potential ansatz from Preprint No. 1882 for the electric fields in the before mentioned electromagnetic scattering problem from single to N profile scattering yields an equivalent system of N integral equations. We present a uniqueness and two existence results for this system depending on the values of the electromagnetic material parameters of the considered biperiodic scatterer. This in particular includes the proof that the system of integral equations is of zero Fredholm index. The general case that the grating interfaces are of polyhedral Lipschitz regularity requires more strict assumptions than the special case of smooth grating interfaces. We exploit the solvability results of this work in a subsequent paper featuring a recursive integral equation algorithm for the 2π-biperiodic multilayered electromagnetic scattering problem.