  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 $\varepsilon>0$.
