Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data

In this paper we investigate vector-valued parabolic initial boundary value problems (A(t, x, D), B-j(t, x, D)) subject to general boundary conditions in domains G in R-n with compact C-2m-boundary. The top-order coefficients of A are assumed to be continuous. We characterize optimal L-p-L-q-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on A and the Lopatinskii-Shapiro condition on (A, B-1 , . . . , B-m) are necessary for these L-p-L-q-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.