Homogenization of elliptic boundary value problems in Lipschitz domains

Carlos E. Kenig, Zhongwei Shen

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

In this paper we study the Lp boundary value problems for L(u)=0 in Rd+1+, where L=-div(A {down triangle, open}) 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 ∞ (Dahlberg's original unpublished proof is given in the Appendix).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 Rd+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.

Original languageEnglish
Pages (from-to)867-917
Number of pages51
JournalMathematische Annalen
Volume350
Issue number4
DOIs
StatePublished - Aug 2011

Bibliographical note

Funding Information:
C. E. Kenig was supported in part by NSF grant DMS-0456583 and Z. Shen was supported in part by NSF grant DMS-0855294.

Keywords

  • 35J25

ASJC Scopus subject areas

  • Mathematics (all)

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