Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

Journal article (2021)

Authors

Felix Hummel Technische Universität München
N. Lindemulder Analysis - EEMCS , Karlsruhe Institut für Technologie
Research Group
Analysis (EEMCS) (TU Delft)
Bessel potential Sobolev Triebel-Lizorkin UMD Weight Maximal regularity Smoothing Vector-valued Mixed-norm Boundary value problem Anistropic Lopatinskii-Shapiro Poisson operator
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Published Date

2021

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A∞-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.