Compactness via symmetrization
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.
scientific article; zbMATH DE number 2107266
scientific article