We first discuss a class of inequalities of Onofri 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 Caffarelli-Kohn-Nirenberg inequality, 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 and a careful analysis of the convergence to a solution of a Lionville equation. In this way, the Onofri inequality appears as a limit case. of the Caffarelli-Kohn-Nirenberg inequality.

Dolbeault, J., Esteban, M., Tarantello, G. (2008). The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, 7(2), 313-341.

The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions

TARANTELLO, GABRIELLA
2008-01-01

Abstract

We first discuss a class of inequalities of Onofri 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 Caffarelli-Kohn-Nirenberg inequality, 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 and a careful analysis of the convergence to a solution of a Lionville equation. In this way, the Onofri inequality appears as a limit case. of the Caffarelli-Kohn-Nirenberg inequality.
2008
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
SOBOLEV INEQUALITIES; ELLIPTIC-EQUATIONS; SHARP; TRUDINGER; BEHAVIOR
Dolbeault, J., Esteban, M., Tarantello, G. (2008). The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, 7(2), 313-341.
Dolbeault, J; Esteban, M; Tarantello, G
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/29102
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