The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions
Dolbeault, Jean; Esteban, Maria J.; Tarantello, Gabriella (2008), The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 7, 2, p. 313–341. http://dx.doi.org/10.2422/2036-2145.2008.2.05
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00139062/en/Date
2008Journal name
Annali della Scuola Normale Superiore di Pisa. Classe di ScienzeVolume
7Number
2Publisher
Le Edizioni della Normale
Pages
313–341
Publication identifier
Metadata
Show full item recordAbstract (EN)
We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it holds if and only if the parameter is in the interval (-1,0]. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy-Sobolev inequality, as established by Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method. In this way, the weighted Moser-Trudinger inequality appears as a limit case of the Hardy-Sobolev inequality.Subjects / Keywords
Weighted Moser-Trudinger inequality; Hardy-Sobolev inequality; Onofri's inequality; Caffarelli-Kohn-Nirenberg inequality; extremal functions; Kelvin transformation; Emden-Fowler transformation; stereographic projection; radial symmetry; symmetry breaking; blow-up analysisRelated items
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Dolbeault, Jean; Esteban, Maria J.; Loss, Michael (2016) Document de travail / Working paper
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