The Neumann problem in Lp on Lipschitz and convex domains

Aekyoung Shin Kim, Zhongwei Shen

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Given p > 2 and a bounded Lipschitz domain Ω, we obtain a necessary and sufficient condition for the solvability of the Neumann problem for Laplace's equation Δ u = 0 in Ω with boundary data in Lp (∂ Ω). As a result, we show that the Lp Neumann problem on convex domains in Rd is solvable for 1 < p < ∞ if d = 2, for 1 < p < 4 if d = 3, and for 1 < p < 3 + ε if d ≥ 4.

Original languageEnglish
Pages (from-to)1817-1830
Number of pages14
JournalJournal of Functional Analysis
Volume255
Issue number7
DOIs
StatePublished - Oct 1 2008

Bibliographical note

Funding Information:
The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: zshen2@email.uky.edu (Z. Shen).

Keywords

  • Convex domain
  • Lipschitz domain
  • Neumann problem

ASJC Scopus subject areas

  • Analysis

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