Elsevier

Journal of Functional Analysis

Volume 255, Issue 7, 1 October 2008, Pages 1817-1830
Journal of Functional Analysis

The Neumann problem in Lp on Lipschitz and convex domains

https://doi.org/10.1016/j.jfa.2008.06.032Get rights and content
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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 Δu=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 d4.

Keywords

Neumann problem
Lipschitz domain
Convex domain

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The authors are supported in part by the NSF (DMS-0500257).