# Maximal Regularity with Weights for Parabolic Problems with Inhomogeneous Boundary Conditions

## TL;DR

This paper proves weighted maximal regularity results for linear parabolic problems with inhomogeneous boundary conditions, using advanced function spaces to improve regularity and avoid boundary compatibility conditions.

## Contribution

It introduces a novel approach employing weighted anisotropic mixed-norm Banach spaces for maximal regularity in parabolic problems with inhomogeneous boundary conditions.

## Key findings

- Established weighted $L^{q}$-$L^{p}$-maximal regularity for parabolic problems.
- Developed trace theory for complex function spaces.
- Avoided boundary compatibility conditions through weighted spaces.

## Abstract

In this paper we establish weighted $L^{q}$-$L^{p}$-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin and Besov type, whose trace theory is also subject of study.