The Neumann problem in Lp on Lipschitz and convex domains

Aekyoung Shin Kim; Zhongwei Shen

doi:10.1016/j.jfa.2008.06.032

The Neumann problem in L^p on Lipschitz and convex domains

Aekyoung Shin Kim, Zhongwei Shen

Research output: Contribution to journal › Article › peer-review

18 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 L^p (∂ Ω). As a result, we show that the L^p Neumann problem on convex domains in R^d 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 language	English
Pages (from-to)	1817-1830
Number of pages	14
Journal	Journal of Functional Analysis
Volume	255
Issue number	7
DOIs	https://doi.org/10.1016/j.jfa.2008.06.032
State	Published - Oct 1 2008

Bibliographical note

Funding Information:
The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: [email protected] (Z. Shen).

Keywords

Convex domain
Lipschitz domain
Neumann problem

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2008.06.032

Cite this

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title = "The Neumann problem in Lp on Lipschitz and convex domains",

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.",

keywords = "Convex domain, Lipschitz domain, Neumann problem",

author = "Kim, {Aekyoung Shin} and Zhongwei Shen",

note = "Funding Information: The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: [email protected] (Z. Shen).",

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AU - Kim, Aekyoung Shin

AU - Shen, Zhongwei

N1 - Funding Information: The authors are supported in part by the NSF (DMS-0500257). Corresponding author. E-mail address: [email protected] (Z. Shen).

PY - 2008/10/1

Y1 - 2008/10/1

N2 - 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.

AB - 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.

KW - Convex domain

KW - Lipschitz domain

KW - Neumann problem

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