Optimal Lp-Lq-regularity for parabolic problems with inhomogeneous boundary data

In this paper we investigate vector-valued parabolic initial boundary value problems (A(t, x,D),Bj(t, x,D)) subject to general boundary conditions in domains G in R with compact C-boundary. The top-order coefficients of A are assumed to be continuous. We characterize optimal L-L-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,B1, . . . ,Bm) are necessary for these L-L-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.