Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces

Abstract

This text is devoted to maximal regularity results for second-order parabolic systems on Lipschitz domains of space dimension $n \ge 3$ with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial-value problems generates isomorphisms between two scales of Sobolev--Morrey spaces for solutions and right-hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower-order coefficients.