The inhomogeneous Dirichlet problem in Lipschitz domains
type: Item
instance of: scholarly article
title: The inhomogeneous Dirichlet problem in Lipschitz domains
zbMATH Open document ID: 0832.35034
author: C. E. Kenig
author: David Jerison
published in: Journal of Functional Analysis
publication date: 1996-02-26T00:00:00Z
review text: The inhomogeneous Dirichlet problem \[ \Delta u= f\qquad\text{on}\quad \Omega,\qquad u= 0\qquad\text{on}\quad \partial\Omega, \] with data in Sobolev spaces of domains \(\Omega\) in \(\mathbb{R}^n\) with Lipschitz boundary, is studied. There are certain special difficulties in this kind of domains, e.g. the customary approach of reducing the inhomogeneous problem to a homogeneous one, in the case of smooth domains, can fail in the case of Lipschitz domains for certain exceptional function spaces. In this work, the authors present a complete description of all Sobolev spaces for which estimates hold, and the best possible estimates in Besov spaces. Some counterexamples in borderline cases are also constructed.
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 764042
zbMATH Keywords: Besov spaces
zbMATH Keywords: counterexamples
reviewed by: Ding Hua
MaRDI profile type: Publication
full work available at URL: jfan.1995.1067
OpenAlex ID: W1965332272
DOI: 10.1006/JFAN.1995.1067
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description: scientific article; zbMATH DE number 764042