On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Loredana Lanzani; Zhongwei Shen

doi:10.1081/pde-120028845

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Loredana Lanzani, Zhongwei Shen

Producción científica: Article › revisión exhaustiva

51 Citas (Scopus)

Resumen

Let Ω be a bounded Lipschitz domain in Rⁿ, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ L^p(∂Ω) on ∂Ω for Laplace's equation δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ε, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ∥(∇u)*∥_p ≤ C∥f∥_p, as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

Idioma original	English
Páginas (desde-hasta)	91-109
Número de páginas	19
Publicación	Communications in Partial Differential Equations
Volumen	29
N.º	1-2
DOI	https://doi.org/10.1081/pde-120028845
Estado	Published - 2005

Nota bibliográfica

Funding Information:
First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.

Financiación

First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.

Financiadores	Número del financiador
National Science Foundation (NSF)	DMS-9800794, DMS-9732894

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1081/pde-120028845

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title = "On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains",

abstract = "Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂Ω) on ∂Ω for Laplace's equation δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ε, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ∥(∇u)*∥p ≤ C∥f∥p, as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.",

keywords = "Laplace's equation, Lipschitz domains, Robin boundary condition",

author = "Loredana Lanzani and Zhongwei Shen",

note = "Funding Information: First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.",

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T1 - On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

AU - Lanzani, Loredana

AU - Shen, Zhongwei

N1 - Funding Information: First author was supported by NSF Grant No. DMS-9800794. Second author was supported by NSF Grant No. DMS-9732894.

PY - 2005

Y1 - 2005

N2 - Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂Ω) on ∂Ω for Laplace's equation δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ε, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ∥(∇u)*∥p ≤ C∥f∥p, as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

AB - Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂Ω) on ∂Ω for Laplace's equation δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ε, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ∥(∇u)*∥p ≤ C∥f∥p, as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

KW - Laplace's equation

KW - Lipschitz domains

KW - Robin boundary condition

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JO - Communications in Partial Differential Equations

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On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

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ASJC Scopus subject areas

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