Parabolic Initial-Boundary Value Problems with Inhomogeneous Data A Maximal Weighted Regularity Approach

Abstract

We develop a new function space theoretic weighted $L^{q}$-$L^{p}$-maximal regularity approach 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. The novelty of our approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin, and Besov type. The main tools are maximal functions and Fourier multipliers, of which we also give a detailed treatment.