The Neumann problem and Helmholtz decomposition in convex domains

Jun Geng; Zhongwei Shen

doi:10.1016/j.jfa.2010.07.005

The Neumann problem and Helmholtz decomposition in convex domains

Jun Geng, Zhongwei Shen

Mathematics

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

35 Scopus citations

Abstract

We show that the Neumann problem for Laplace's equation in a convex domain ω with boundary data in L^p(∂ω) is uniquely solvable for 1 < p < ∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in L^p(ω,R^d).

Original language	English
Pages (from-to)	2147-2164
Number of pages	18
Journal	Journal of Functional Analysis
Volume	259
Issue number	8
DOIs	https://doi.org/10.1016/j.jfa.2010.07.005
State	Published - Oct 2010

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: jgeng@ms.uky.edu (J. Geng), zshen2@email.uky.edu (Z. Shen). 1 Supported in part by NSF grant DMS-0855294.

Keywords

Convex domains
Helmholtz decomposition
Neumann problem

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2010.07.005

Cite this

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title = "The Neumann problem and Helmholtz decomposition in convex domains",

abstract = "We show that the Neumann problem for Laplace's equation in a convex domain ω with boundary data in Lp(∂ω) is uniquely solvable for 1 < p < ∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(ω,Rd).",

keywords = "Convex domains, Helmholtz decomposition, Neumann problem",

author = "Jun Geng and Zhongwei Shen",

note = "Funding Information: * Corresponding author. E-mail addresses: jgeng@ms.uky.edu (J. Geng), zshen2@email.uky.edu (Z. Shen). 1 Supported in part by NSF grant DMS-0855294.",

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AU - Geng, Jun

AU - Shen, Zhongwei

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PY - 2010/10

Y1 - 2010/10

N2 - We show that the Neumann problem for Laplace's equation in a convex domain ω with boundary data in Lp(∂ω) is uniquely solvable for 1 < p < ∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(ω,Rd).

AB - We show that the Neumann problem for Laplace's equation in a convex domain ω with boundary data in Lp(∂ω) is uniquely solvable for 1 < p < ∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(ω,Rd).

KW - Convex domains

KW - Helmholtz decomposition

KW - Neumann problem

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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ER -

The Neumann problem and Helmholtz decomposition in convex domains

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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