Towards an L-theory for Vector-valued Elliptic Boundary Value Problems

Vector-valued elliptic and parabolic boundary value problems subject to general boundary conditions have been investigated recently in [DHP01] in the Lp-context for 1 < p < 1. One of the main goals of this paper was to deduce a maximal Lp-regularity result for the solution of the parabolic initial boundary value problem. A classical reference in the elliptic context are the celebrated papers of Agmon, Douglis and Nirenberg [ADN59]. For further references and information on the scalar and vector-valued case we refer to the [Ama01] and the list of references given in [DHP01]. Vector-valued elliptic and parabolic problems on all of R we considered rst by Amann [Ama01] on a large scale of function spaces, including L1(R n ;E). Here E denotes an arbitrary Banach space. He proved in particular that the L1-realization of such problems generates an analytic C0-semigroup provided the top-order coeÆcients of the underlying operators are uniformly bounded and H older continuous. In this note, we consider vector-valued boundary value problems with constant coeÆcients in the L1-setting for a half space. Following the approach described in [DHP01], we assume the Lopatinskii-Shapiro to be true; we then obtain a representation of the solution u of the elliptic problem by integral operators which allows to deduce a-priori estimates for u in the L1(R n+1 + ;E)-norm. Here E denotes again an arbitrary Banach space. These estimates imply in particular that the L1(R n+1 + ;E)-realization of an elliptic boundary value problem with constant coeÆcients in the half space R + generates an analytic C0-semigroups on L1(R n+1 + ;E). For di erent approaches and results with variable coeÆcients in the scalar-valued case we refer to Amann [Ama83], Di Blasio [DiB91], Guidetti [Gui93] and Tanabe [Tan97], Section 5.4.