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 del) 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+ delta. We also present a new proof of Dahlberg's theorem on the L-p Dirichlet problem for 2 - delta < p < infinity (Dahlberg's original unpublished proof is given in the Appendix).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.