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

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
External document link
http://hal.archives-ouvertes.fr/hal-00496936/fr/
Date
2012
Journal name
Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences
Volume
142
Number
4
Publisher
Cambridge University Press
Pages
745-767
Publication identifier
http://dx.doi.org/10.1017/S0308210510001101
Metadata
Show full item record
Author(s)
Dolbeault, Jean cc
Esteban, Maria J. cc
Abstract (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.
Subjects / Keywords
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

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