Dirichlet and Neumann boundary conditions: what is in between?
type: Item
instance of: scholarly article
title: Dirichlet and Neumann boundary conditions: what is in between?
zbMATH Open document ID: 1044.31005
author: Wolfgang Arendt
author: Mahamadi Warma
published in: Journal of Evolution Equations
publication date: 2003-07-29T00:00:00Z
review text: Given an open set \(\Omega\) in \(\mathbb{R}^n\), an admissible measure on \(\partial \Omega\) is a Radon measure \(\mu\) on the Borel \(\sigma\)-field of some open subset \(\Gamma_{\mu}\) of \(\partial \Omega\) which does not charge sets of capacity zero. For any admissible measure \(\mu\) on \(\partial \Omega\), the authors define a realization \(\Delta_{\mu}\) of the Laplacian in \(L^2(\Omega)\) with general Robin boundary conditions and show that \(\Delta_{\mu}\) generates a holomorphic \(C_0\)-semigroup on \(L^2(\Omega)\) which is sandwiched between the Dirichlet Laplacian and the Neumann Laplacian semigroups. Furthermore, the authors show that, under some locality and regularity assumptions, the generator of any sandwiched semigroup is of the form \(\Delta_{\mu}\). The authors also show that if \(D(\Delta_{\mu})\) contains smooth functions, then \(\mu\) is absolutely continuous with respect to the \((n-1)\)-dimensional Hausdorff measure on \(\Omega\).
reviewed by: Renming Song
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1921151
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: Robin boundary conditions
zbMATH Keywords: holomorphic \(C_0\)-semigroup
zbMATH Keywords: Dirichlet Laplacian
zbMATH Keywords: Neumann Laplacian
MaRDI profile type: Publication
full work available at URL: s000280300005
OpenAlex ID: W4243998800
DOI: 10.1007/S000280300005
description: scientific article; zbMATH DE number 1921151