Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

Dolbeault, Jean; Esteban, Maria J.

Dolbeault, Jean; Esteban, Maria J. (2012), Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences, 142, 4, p. 745-767. http://dx.doi.org/10.1017/S0308210510001101

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00496936/fr/

Date

2012

Nom de la revue

Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences

Volume

142

Numéro

Éditeur

Cambridge University Press

Pages

745-767

Résumé (EN)

We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities which have been obtained recently as a limit case of the first ones. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo-Nirenberg interpolation inequalities and Gross' logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.

Mots-clés

Caffarelli-Kohn-Nirenberg inequality; Hardy-Sobolev inequality; Sobolev spaces; existence; symmetry breaking; compactness; radial symmetry; Emden-Fowler transformation; Kelvin transformation; extremal functions; logarithmic Hardy inequality