Mathematische Annalen

, Volume 350, Issue 4, pp 867–917

Homogenization of elliptic boundary value problems in Lipschitz domains



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 ∞ (Dahlberg’s original unpublished proof is given in the Appendix).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)



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© Springer-Verlag 2010

Authors and Affiliations

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

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