Uniform concentration-compactness for Sobolev spaces on variable domains
type: Item
instance of: scholarly article
title: Uniform concentration-compactness for Sobolev spaces on variable domains
zbMATH Open document ID: 0957.49027
author: Dorin Bucur
published in: Journal of Differential Equations
publication date: 2001-03-11T00:00:00Z
review text: The author presents a new method for proving existence results for some classes of shape optimization problems involving the eigenvalues of Laplace operator with Dirichlet homogeneous boundary conditions. Such method exploits the concentration-compactness principle, together with the \(\gamma\)-convergence theory of sets. The main result of the paper is expressed as a concentration-compactness theorem in \({\mathcal L}(L^2({\mathbb R}^N))\) for the sequence of the resolvent operators \(\{R_{A_n}\}_{n\in{\mathbb N}}\) of the Laplace operator with Dirichlet homogeneous boundary conditions, where \(\{A_n\}_{n\in{\mathbb N}}\) is a sequence of open (or quasi open) sets with uniformly bounded Lebesgue measure. The analysis of the asymptotic behaviour of \(\{A_n\}_{n\in{\mathbb N}}\), in the sense of \(\gamma\)-convergence, and of \(\{R_{A_n}\}_{n\in{\mathbb N}}\), in the sense of the uniform operator topology of \(L^2({\mathbb R}^N)\), is carried out, and the various possible situations are described. The results are then applied for proving existence results of global minimizers in sets of quasi open sets, not necessarily bounded by a uniform bound, but just with uniformly bounded measures, for a class of shape optimization problems depending on the eigenvalues of Laplace operator with Dirichlet homogeneous boundary conditions.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1454740
zbMATH Keywords: concentration-compactness principle
zbMATH Keywords: gamma convergence
zbMATH Keywords: resolvent operators
zbMATH Keywords: shape optimization
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3726
OpenAlex ID: W2087025557
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DOI: 10.1006/JDEQ.1999.3726
description: scientific article; zbMATH DE number 1454740