Mathematische Annalen

, Volume 350, Issue 4, pp 867–917

Homogenization of elliptic boundary value problems in Lipschitz domains

Article

Abstract

In this paper we study the Lp boundary value problems for \({\mathcal{L}(u)=0}\) in \({\mathbb{R}^{d+1}_+}\) , where \({\mathcal{L}=-{\rm div} (A\nabla )}\) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in xd+1 and satisfies some minimal smoothness condition in the xd+1 variable, we show that the Lp Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the Lp Dirichlet problem for 2 − δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the xd+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform Lp estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.

Mathematics Subject Classification (2000)

35J25 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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