The Lp boundary value problems on Lipschitz domains

Zhongwei Shen

doi:10.1016/j.aim.2007.05.017

The L^p boundary value problems on Lipschitz domains

Zhongwei Shen

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

82 Scopus citations

Abstract

Let Ω be a bounded Lipschitz domain in Rⁿ. We develop a new approach to the invertibility on L^p (∂ Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and frac(2 (n - 1), (n + 1)) - ε < p < 2 where ε > 0 depends on Ω, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.

Original language	English
Pages (from-to)	212-254
Number of pages	43
Journal	Advances in Mathematics
Volume	216
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2007.05.017
State	Published - Dec 1 2007

Bibliographical note

Funding Information:
1 The author is supported in part by the NSF (DMS-0500257).

Funding

1 The author is supported in part by the NSF (DMS-0500257).

Funders	Funder number
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
Fundação para a Ciência e Tecnologia I.P.	PTDC/CCI-BIO/29266/2017
Fundação para a Ciência e Tecnologia I.P.

Keywords

Biharmonic equation
Dirichlet problem
Elliptic systems
Lipschitz domains
Neumann problem

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1016/j.aim.2007.05.017

Cite this

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title = "The Lp boundary value problems on Lipschitz domains",

abstract = "Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂ Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and frac(2 (n - 1), (n + 1)) - ε < p < 2 where ε > 0 depends on Ω, we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.",

keywords = "Biharmonic equation, Dirichlet problem, Elliptic systems, Lipschitz domains, Neumann problem",

author = "Zhongwei Shen",

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T1 - The Lp boundary value problems on Lipschitz domains

AU - Shen, Zhongwei

N1 - Funding Information: 1 The author is supported in part by the NSF (DMS-0500257).

PY - 2007/12/1

Y1 - 2007/12/1

N2 - Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂ Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and frac(2 (n - 1), (n + 1)) - ε < p < 2 where ε > 0 depends on Ω, we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.

AB - Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂ Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and frac(2 (n - 1), (n + 1)) - ε < p < 2 where ε > 0 depends on Ω, we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.

KW - Biharmonic equation

KW - Dirichlet problem

KW - Elliptic systems

KW - Lipschitz domains

KW - Neumann problem

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