Domain perturbation for elliptic equations subject to Robin boundary conditions
type: Item
instance of: scholarly article
title: Domain perturbation for elliptic equations subject to Robin boundary conditions
zbMATH Open document ID: 0886.35063
published in: Journal of Differential Equations
publication date: 1998-05-10T00:00:00Z
review text: The aim of this very important paper is to study how the solutions of the linear or nonlinear equation \[ -\Delta u=f(u) \quad\text{in }\Omega_n, \qquad \frac{\partial u}{\partial\nu}+ \beta_0 u=0 \quad\text{on }\partial\Omega_n \] behave as the domains \(\Omega_n\) approach an open bounded set \(\Omega\), where \(\Delta\) is the Laplace operator, \(\nu\) the outer unit normal to the boundary \(\partial\Omega_n\) of \(\Omega_n\) and \(\beta_0>0\) a constant. The boundary conditions in the limit may be different from the original ones (this depends on how the domains approach \(\Omega\)). Main result: The authors present the proof of the domain perturbation theorems for linear and nonlinear elliptic equations under Robin boundary conditions. The theory allows very singular perturbation of domains. In particular, it includes cutting holes, parts degenerating to a set of measure zero such as the dumbbell problem, or wildly oscillating boundaries. One proves, in the case of wildly oscillating boundaries, that the limiting problem is the Dirichlet problem.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1054116
zbMATH Keywords: limiting problem
zbMATH Keywords: domain perturbation theorems
zbMATH Keywords: cutting holes
zbMATH Keywords: Dumbbell problem
zbMATH Keywords: wildly oscillating boundaries
author: E. N. Dancer
author: Daniel Daners
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3256
OpenAlex ID: W2085203688
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DOI: 10.1006/JDEQ.1997.3256
description: scientific article; zbMATH DE number 1054116