Homogenization of elliptic boundary value problems in Lipschitz domains

Abstract

In this paper we study the L ^p boundary value problems for ${\mathcal{L}(u)=0}$ in ${\mathbb{R}^{d+1}_+}$ , where ${\mathcal{L}=-{\rm div} (A\nabla )}$ is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x _d+1 and satisfies some minimal smoothness condition in the x _d+1 variable, we show that the L ^p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L ^p Dirichlet problem for 2 − δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x _d+1 variable, these results extend directly from ${\mathbb{R}^{d+1}_+}$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L ^p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.