# Homogenization of elliptic problems with*L*^{p} boundary data

Article

- Accepted:

DOI: 10.1007/BF01442648

- Cite this article as:
- Avellaneda, M. & Lin, FH. Appl Math Optim (1987) 15: 93. doi:10.1007/BF01442648

## Abstract

We consider the homogenization problem where

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

*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*_{ε}*converges 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.## Copyright information

© Springer-Verlag New York Inc. 1987