Schauder and LPEstimates for Parabolic systems via campanato spaces

Abstract

The following note deals with classical Schauder and L{sup p} estimates in the setting of parabolic systems. For the heat equation these estimates are usually obtained via potential theoretic methods, i.e., by studying the fundamental solution, and for the elliptic case. For systems, however, it has become customary to base both Schauder and L{sup p} theory on Campanato`s technique. For the elliptic case this is explained in and (see also the references therein to Campanato`s original papers) and for the stationary Stokes system in. The purpose of this note is to show how this technique can be applied to parabolic systems. In many ways it is very similar to Giaquinta`s exposition-in spirit as well as in detail. However, the parabolic case dose offer certain peculiarities, most notably the global Schauder estimates, where we have to deal with a compatibility condition on the data. 13 refs.