Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions

TL;DR

This paper introduces a maximal regularity framework in weighted $L_p$-spaces for parabolic boundary value problems, enabling reduced initial regularity requirements and boundary compatibility conditions, with applications to static and relaxation types.

Contribution

It develops a novel weighted maximal regularity approach for parabolic problems with inhomogeneous boundary conditions, utilizing advanced interpolation, trace theory, and operator calculus.

Findings

Reduces initial regularity requirements for solutions.

Avoids boundary compatibility conditions due to weighted framework.

Provides smoothing effects for solutions in weighted spaces.

Abstract

We develop a maximal regularity approach in temporally weighted $L_p$-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal weights, operator-valued functional calculus, as well as localization and perturbation arguments.