Applied Mathematics and Optimization

, Volume 15, Issue 1, pp 93–107

Homogenization of elliptic problems withLp boundary data

  • Marco Avellaneda
  • Fang-Hua Lin
Article

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 aC1,α, periodic, uniformly positive matrix, and the datag belongs toLp(∂D), 1 <p < ∞. We show that, if∂D satisfies a uniform exterior sphere condition, thenuεconverges in Lp(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.

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Marco Avellaneda
    • 1
  • Fang-Hua Lin
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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