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 publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00496936/fr/Date
2012Journal name
Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical SciencesVolume
142Number
4Publisher
Cambridge University Press
Pages
745-767
Publication identifier
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Show full item recordAbstract (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 inequalityRelated items
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