Sobolev-Morrey spaces associated with evolution equations

Abstract

In this text we introduce new classes of Sobolev--Morrey spaces being adequate for the regularity theory of second-order parabolic boundary-value problems on Lipschitz domains of space dimension $n \ge 3$ with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, Lipschitz transformation, and reflection. In the second part [11] of our presentation we show that the class of second-order parabolic systems with diagonal principal part generates isomorphisms between the above-mentioned Sobolev--Morrey spaces of solutions and right-hand sides.