Second Order Operators Subject to Dirichlet Boundary Conditions in Weighted Triebel-Lizorkin Spaces: Parabolic Problems

TL;DR

This paper develops a framework for analyzing second order parabolic PDEs with Dirichlet boundary conditions using weighted Triebel-Lizorkin spaces, enabling treatment of rough boundary data and improved regularity results.

Contribution

It introduces weighted $L_q$-maximal regularity results for parabolic problems in Triebel-Lizorkin spaces, accommodating rough boundary data without compatibility conditions.

Findings

Established weighted maximal regularity in Triebel-Lizorkin spaces.

Allowed for rough inhomogeneous boundary data.

Provided smoothing effects and flexibility in data regularity.

Abstract

In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.