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 ℝn 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, B1... Bm) 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.
ASJC Scopus subject areas
- 数学 (全般)