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

Producción científica: Article › revisión exhaustiva

21 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	1817-1830
Número de páginas	14
Publicación	Journal of Functional Analysis
Volumen	255
N.º	7
DOI	https://doi.org/10.1016/j.jfa.2008.06.032
Estado	Published - oct 1 2008

Nota bibliográfica

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

Financiación

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

Financiadores	Número del financiador
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-0500257
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China

ASJC Scopus subject areas

Analysis

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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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