Unique solvability of the Neumann problem with weighted boundary data on a bounded C1 domain

Yong Ding; Xudong Lai

doi:10.1016/j.jmaa.2016.06.034

Unique solvability of the Neumann problem with weighted boundary data on a bounded C¹ domain

Yong Ding
, Xudong Lai^*

^*Corresponding author for this work

Beijing Normal University

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

1 Scopus citations

Abstract

Let D⊂Rⁿ(n≥3) be a bounded C¹ domain. We prove that if 1>p>∞ and w∈A_p(∂D), then the following Neumann problem {Δu=0,in D=g,on ∂D, has a unique (up to a constant) solution u with the boundary data g∈L^p(∂D,w) satisfying ∫_∂Dg(Q)dσ(Q)=0. Moreover, u satisfies‖(∇u)_⁎,α‖_L^p(∂D,w)≤C‖g‖_L^p(∂D,w) for each 0>α>1, where (∇u)_⁎,α denotes the nontangential maximal function of ∇u.

Original language	English
Pages (from-to)	340-369
Number of pages	30
Journal	Journal of Mathematical Analysis and Applications
Volume	444
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2016.06.034
State	Published - 1 Dec 2016
Externally published	Yes

Keywords

C domain
Laplace's equation
The Neumann problem
Weighted spaces

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title = "Unique solvability of the Neumann problem with weighted boundary data on a bounded C1 domain",

abstract = "Let D⊂Rn(n≥3) be a bounded C1 domain. We prove that if 1>p>∞ and w∈Ap(∂D), then the following Neumann problem \{Δu=0,in D=g,on ∂D, has a unique (up to a constant) solution u with the boundary data g∈Lp(∂D,w) satisfying ∫∂Dg(Q)dσ(Q)=0. Moreover, u satisfies‖(∇u)⁎,α‖Lp(∂D,w)≤C‖g‖Lp(∂D,w) for each 0>α>1, where (∇u)⁎,α denotes the nontangential maximal function of ∇u.",

keywords = "C domain, Laplace's equation, The Neumann problem, Weighted spaces",

author = "Yong Ding and Xudong Lai",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2016",

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doi = "10.1016/j.jmaa.2016.06.034",

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TY - JOUR

T1 - Unique solvability of the Neumann problem with weighted boundary data on a bounded C1 domain

AU - Ding, Yong

AU - Lai, Xudong

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Y1 - 2016/12/1

N2 - Let D⊂Rn(n≥3) be a bounded C1 domain. We prove that if 1>p>∞ and w∈Ap(∂D), then the following Neumann problem {Δu=0,in D=g,on ∂D, has a unique (up to a constant) solution u with the boundary data g∈Lp(∂D,w) satisfying ∫∂Dg(Q)dσ(Q)=0. Moreover, u satisfies‖(∇u)⁎,α‖Lp(∂D,w)≤C‖g‖Lp(∂D,w) for each 0>α>1, where (∇u)⁎,α denotes the nontangential maximal function of ∇u.

AB - Let D⊂Rn(n≥3) be a bounded C1 domain. We prove that if 1>p>∞ and w∈Ap(∂D), then the following Neumann problem {Δu=0,in D=g,on ∂D, has a unique (up to a constant) solution u with the boundary data g∈Lp(∂D,w) satisfying ∫∂Dg(Q)dσ(Q)=0. Moreover, u satisfies‖(∇u)⁎,α‖Lp(∂D,w)≤C‖g‖Lp(∂D,w) for each 0>α>1, where (∇u)⁎,α denotes the nontangential maximal function of ∇u.

KW - C domain

KW - Laplace's equation

KW - The Neumann problem

KW - Weighted spaces

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U2 - 10.1016/j.jmaa.2016.06.034

DO - 10.1016/j.jmaa.2016.06.034

M3 - 文章

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SN - 0022-247X

VL - 444

SP - 340

EP - 369

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Unique solvability of the Neumann problem with weighted boundary data on a bounded C¹ domain

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