Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L^p boundary value problems for L(u)=0 in R^d+1₊, where L=-div(A {down triangle, open}) 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 ∞ (Dahlberg's original unpublished proof is given in the Appendix).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 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.