## Abstract

In this paper we study the L^{p} boundary value problems for L(u)=0 in R^{d+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 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 ∞ (Dahlberg's original unpublished proof is given in the Appendix).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 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.

Original language | English |
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Pages (from-to) | 867-917 |

Number of pages | 51 |

Journal | Mathematische Annalen |

Volume | 350 |

Issue number | 4 |

DOIs | |

State | Published - 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)