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Mathematische Annalen

, Volume 350, Issue 4, pp 867–917 | Cite as

Homogenization of elliptic boundary value problems in Lipschitz domains

  • Carlos E. Kenig
  • Zhongwei ShenEmail author
Article

Abstract

In this paper we study the L p 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 x d+1 and satisfies some minimal smoothness condition in the x d+1 variable, we show that the L p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L p 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 ∞ (Dahlberg’s original unpublished proof is given in the Appendix).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 x d+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L p 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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