The mixed problem in L p for some two-dimensional Lipschitz domains

Loredana Lanzani; Luca Capogna; Russell M. Brown

doi:10.1007/s00208-008-0223-6

The mixed problem in L ^p for some two-dimensional Lipschitz domains

Loredana Lanzani, Luca Capogna, Russell M. Brown

Department of Mathematics

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

14 Scopus citations

Abstract

We consider the mixed problem, {Δ u = 0 in Ω ∂u = f _N on N u = f_D on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f _D , has one derivative in L ^p (D) of the boundary and the Neumann data, f _N , is in L ^p (N). We find a p ₀ > 1 so that for p in an interval (1, p ₀), we may find a unique solution to the mixed problem and the gradient of the solution lies in L ^p .

Original language	English (US)
Pages (from-to)	91-124
Number of pages	34
Journal	Mathematische Annalen
Volume	342
Issue number	1
DOIs	https://doi.org/10.1007/s00208-008-0223-6
State	Published - Sep 2008

ASJC Scopus subject areas

Mathematics(all)

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abstract = "We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .",

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N2 - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

AB - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

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