Homogenization of elliptic problems with L^p boundary data

We consider the homogenization problem {Mathematical expression} where D is a bounded domain, a is a C^1,α, periodic, uniformly positive matrix, and the data g belongs to L^p(∂D), 1 <p < ∞. We show that, if ∂D satisfies a uniform exterior sphere condition, then u_εconverges in L^p(D) to the solution of the corresponding homogenized problem as ε → 0. The proof is done via estimates for Poisson's kernel. We give examples showing that this convergence result does not hold for a general G-convergent sequence of operators and depends on the periodicity of a as well as on its smoothness.