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

Jean Dolbeault; Maria J. Esteban; Gabriella Tarantello

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 2, page 313-341
  • ISSN: 0391-173X

Abstract

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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 Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn-Nirenberg inequality.

How to cite

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Dolbeault, Jean, Esteban, Maria J., and Tarantello, Gabriella. "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 (2008): 313-341. <http://eudml.org/doc/272292>.

@article{Dolbeault2008,
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 Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn-Nirenberg inequality.},
author = {Dolbeault, Jean, Esteban, Maria J., Tarantello, Gabriella},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {bow up method; Liouville equation},
language = {eng},
number = {2},
pages = {313-341},
publisher = {Scuola Normale Superiore, Pisa},
title = {The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions},
url = {http://eudml.org/doc/272292},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Dolbeault, Jean
AU - Esteban, Maria J.
AU - Tarantello, Gabriella
TI - The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 2
SP - 313
EP - 341
AB - 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 Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn-Nirenberg inequality.
LA - eng
KW - bow up method; Liouville equation
UR - http://eudml.org/doc/272292
ER -

References

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