About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

Esteban, Maria J.; Dolbeault, Jean

Esteban, Maria J.; Dolbeault, Jean (2012), About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities, dans Karlsen, Kenneth H.; Holden, Helge, Nonlinear Partial Differential Equations - The Abel Symposium 2010, Springer : Berlin, p. 117-130. http://dx.doi.org/10.1007/978-3-642-25361-4_6

Type

Chapitre d'ouvrage

Lien vers un document non conservé dans cette base

http://hal.archives-ouvertes.fr/hal-00539019/fr/

Date

2012

Titre de l'ouvrage

Nonlinear Partial Differential Equations - The Abel Symposium 2010

Auteurs de l’ouvrage

Karlsen, Kenneth H.; Holden, Helge

Éditeur

Springer

Titre de la collection

Abel Symposia

Numéro dans la collection

vol.7

Ville d’édition

Berlin

Isbn

978-3-642-25360-7

Nombre de pages

360

Pages

117-130

Résumé (EN)

This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Mots-clés

optimal constants; compactness; existence; linearization; Emden-Fowler transformation; symmetry breaking; radial symmetry; extremal functions; logarithmic Sobolev inequality; logarithmic Hardy inequality; Gagliardo-Nirenberg inequality; Caffarelli-Kohn-Nirenberg inequality