Homogenization of elliptic problems withL ^p boundary data

Abstract

We consider the homogenization problem

$$\begin{gathered} - \frac{\partial }{{\partial x_i }}\left( {a^{ij} \left( {\frac{x}{\varepsilon }} \right)\frac{{\partial u_\varepsilon }}{{\partial x_j }}} \right) = 0inD, \hfill \\ u_\varepsilon = gon\partial D, \hfill \\ \end{gathered} $$

whereD is a bounded domain,a is aC ^1,α, periodic, uniformly positive matrix, and the datag belongs toL ^p (∂D), 1 <p < ∞. We show that, if∂D satisfies a uniform exterior sphere condition, thenu _ε converges in L ^p (D) to the solution of the corresponding homogenized problem asε → 0. The proof is done via estimates for Poisson's kernel. We give examples showing that this convergence result does not hold for a generalG-convergent sequence of operators and depends on the periodicity ofa as well as on its smoothness.