Compactness via symmetrization
type: Item
instance of: scholarly article
title: Compactness via symmetrization
zbMATH Open document ID: 1065.49006
author: Almut Burchard
author: Yan Guo
published in: Journal of Functional Analysis
publication date: 2004-10-12T00:00:00Z
review text: The main goal of the paper under review is to examine the role of translations for minimizing sequences via elementary knowledge of their symmetrizations. Precisely, let \(\mathcal F\) and \(\mathcal J\) be two translation-invariant functionals on \(\mathbb R^m,\) and let \(f_n\) be a functional sequence whose corresponding rearrangements \(f^\ast_n\) are convergent. The authors show that \(f_n\) themselves converge up to translations if either \(\lim_{n\to\infty}{\mathcal F}(f_n)=\lim_{n\to\infty}{\mathcal F}(f_n^\ast)\) or \(\lim_{n\to\infty}{\mathcal J}(f_n)=\lim_{n\to\infty}{\mathcal J}(f_n^\ast).\) Various applications are given to dynamical stability of gaseous stars (compressible Euler--Poisson system), to stability in galactic dynamics (Vlassov--Poisson system) and to Hardy--Littlewood--Sobolev's and Sobolev's inequalities.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 2107266
zbMATH Keywords: symmetric rearrangement
zbMATH Keywords: concentration compactness
zbMATH Keywords: dynamic stability
zbMATH Keywords: Hardy-Littlewood-Sobolev inequality
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: Euler-Poisson system
zbMATH Keywords: Vlasov-Poisson system
zbMATH Keywords: stellar dynamics
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.04.005
OpenAlex ID: W2095194783
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DOI: 10.1016/J.JFA.2004.04.005
description: scientific article; zbMATH DE number 2107266