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 ∞ (Dahlberg’s original unpublished proof is given in the Appendix).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.