The Laplacian with Robin boundary conditions on arbitrary domains
type: Item
instance of: scholarly article
title: The Laplacian with Robin boundary conditions on arbitrary domains
zbMATH Open document ID: 1028.31004
DOI: 10.1023/A:1024181608863
author: Wolfgang Arendt
author: Mahamadi Warma
published in: Potential Analysis
publication date: 2003-08-21T00:00:00Z
review text: The authors consider the Laplace operator \(\Delta\) on an open set \(\Omega\) in \(\mathbb{R}^n\) with the Robin boundary condition. They deal with the case of an arbitrary Borel measure \(\mu\) on the boundary \(\partial \Omega\), and show that it is always possible to define a realization \(\Delta_\mu\) of the Laplace operator on \(L^2(\Omega)\) with generalized Robin boundary condition. The operator \(\Delta_\mu\) generates a sub-Markovian semigroup on \(L^2(\Omega)\) which allows Gaussian estimates according to [\textit{W. Arendt} and \textit{A. F. M. ter Elst}, J. Oper. Theory 38, 87-130 (1997; Zbl 0879.35041)]. In the case \(\mu=\beta d\sigma\), corresponding to \(\frac{\partial u}{\partial\nu}+ \beta u =0\) on \(\partial \Omega\), where \(\beta\) is a strictly positive bounded Borel measurable function on \(\partial \Omega\), \ \(\sigma\) is the \((n-1)\)-dimensional Hausdorff measure on \(\partial \Omega\), the authors show that the semigroup has Gaussian estimates with modified exponents. The case of \((n-1)\)-dimensional Hausdorff measure on \(\partial \Omega\) was previously treated in [\textit{D. Daners}, Trans. Am. Math. Soc. 352, 4207-4236 (2000; Zbl 0947.35072)] (for the case of an elliptic second order differential operator). The authors' approach is alternative and is more based on a capacity approach. The authors also show that the spectrum of the Laplace operator with Robin condition in \(L^p(\Omega), 1\leq p <\infty\), is independent of \(p\).
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1968879
zbMATH Keywords: relative capacity
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: Robin boundary conditions
reviewed by: S. G. Samko
MaRDI profile type: Publication
full work available at URL: a:1024181608863
OpenAlex ID: W154424400
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description: scientific article; zbMATH DE number 1968879