Optimal Lp-Lq-regularity for parabolic problems with inhomogeneous boundary data
R. Denk, M. Hieber, J. Prüss
Abstract. In this paper we investigate vector-valued parabolic initial boundary value problems (A(x,D), B1(x,D),..., Bm(x,D)) subject to general boundary conditions in a domain G with compact boundary. The top-order coefficients of the operator A are assumed to be continuous. We characterize optimal Lp-Lq-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(x,D), B1(x,D),..., Bm(x,D)) are necessary for these Lp-Lq-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.
Math. Z. 257, 193-224 (2007)
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