Layer Potential Methods for Elliptic Homogenization Problems

In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $\mathrm{\Omega }$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the theory of homogenization. Let ${\mathcal{L}}_{\epsilon }=-\text{div}(A({\epsilon }^{-1}X)\mathrm{\nabla })$. 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 ${\mathcal{L}}_{\epsilon }(u_{\epsilon })=0$ in $\mathrm{\Omega }$ with optimal estimates uniform in $\epsilon >0$.