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, in 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

External document link

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

Date

2012

Book title

Nonlinear Partial Differential Equations - The Abel Symposium 2010

Book author

Karlsen, Kenneth H.; Holden, Helge

Publisher

Springer

Series title

Abel Symposia

Series number

vol.7

Published in

Berlin

ISBN

978-3-642-25360-7

Number of pages

360

Pages

117-130

Abstract (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).

Subjects / Keywords

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