Mathematische Artikel [31]

Optimal L_p-L_q-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),..., B_m(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 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(x,D), B₁(x,D),..., B_m(x,D)) 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.

Math. Z. 257, 193-224 (2007)

The paper is available here: pdf