Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains

TL;DR

This paper proves the solvability of boundary value problems for elliptic operators with periodic coefficients in Lipschitz domains, extending classical results and providing new proofs and estimates relevant to homogenization theory.

Contribution

It establishes new solvability results for Neumann and regularity problems in Lipschitz domains with periodic coefficients, and offers a novel proof of Dahlberg's theorem for the Dirichlet problem.

Findings

Unique solvability of $L^p$ Neumann and regularity problems for $1<p<2+ ext{delta}$.

Extension of classical boundary value problem results to Lipschitz domains.

New proof of Dahlberg's theorem on the $L^p$ Dirichlet problem.

Abstract

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is {\it 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 $2-\delta<p< \infty$ (Dahlberg's original unpublished proof is given in the Appendix).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.