Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

TL;DR

This paper advances the understanding of elliptic and parabolic boundary value problems by establishing weighted maximal regularity results in various sophisticated function spaces, accommodating rough boundary data and enhancing regularity analysis.

Contribution

It introduces weighted $L_q$-maximal regularity results in Besov and Triebel-Lizorkin spaces for parabolic problems with power weights, extending beyond traditional $A_p$-range assumptions.

Findings

Solved weighted $L_q$-maximal regularity in Besov and Triebel-Lizorkin spaces.

Extended regularity results to rougher boundary data.

Provided a quantitative smoothing effect inside the domain.

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 $L_{q}$-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty}$-class. In Besov space case we have the restriction that the microscopic parameter equals to $q$. Going beyond the $A_{p}$-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.