Layer Potential Methods for Elliptic Homogenization Problems

TL;DR

This paper employs layer potential methods to analyze boundary value problems for elliptic systems with rapidly oscillating periodic coefficients, establishing uniform solvability and estimates in Lipschitz domains relevant to homogenization theory.

Contribution

It introduces layer potential techniques to prove uniform solvability and estimates for elliptic boundary value problems with oscillating coefficients in Lipschitz domains.

Findings

Established solvability of Dirichlet, regularity, and Neumann problems for all ε>0.

Proved uniform estimates independent of ε for solutions.

Extended layer potential methods to homogenization problems with oscillating coefficients.

Abstract

In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $\Omega$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the theory of homogenization. Let $\mathcal{L}_\varepsilon=-\text{div}\big(A(\varepsilon^{-1}X)\nabla \big)$. Under the assumption that $A(X)$ is elliptic, symmetric, periodic and H\"older continuous, we establish the solvability of the $L^2$ Dirichlet, regularity, and Neumann problems for $\mathcal{L}_\varepsilon (u_\varepsilon)=0$ in $\Omega$ with optimal estimates uniform in $\varepsilon>0$.