Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

• Published in: Journal of evolution equations Vol. 20; no. 1; pp. 59 - 108
• Main Author: Lindemulder, Nick
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2020; Springer Nature B.V
• Subjects: Analysis; Banach spaces; Boundary conditions; Boundary value problems; Function space; Mathematics; Mathematics and Statistics; Regularity; Anisotropic spaces; Parabolic initial-boundary value problems; Bessel potential; Traces; Besov; Maximal regularity; 46E40; Mixed-norms; Inhomogeneous boundary conditions; Primary 35K50; 46E35; 42B25; Weights; Triebel–Lizorkin; Sobolev; Secondary 42B15; Vector-valued
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-019-00515-7

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.