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About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

Esteban, Maria J.; Dolbeault, Jean (2012), About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities, in Karlsen, Kenneth H.; Holden, Helge, Nonlinear Partial Differential Equations - The Abel Symposium 2010, Springer : Berlin, p. 117-130. http://dx.doi.org/10.1007/978-3-642-25361-4_6

Type
Chapitre d'ouvrage
External document link
http://hal.archives-ouvertes.fr/hal-00539019/fr/
Date
2012
Book title
Nonlinear Partial Differential Equations - The Abel Symposium 2010
Book author
Karlsen, Kenneth H.; Holden, Helge
Publisher
Springer
Series title
Abel Symposia
Series number
vol.7
Published in
Berlin
ISBN
978-3-642-25360-7
Number of pages
360
Pages
117-130
Publication identifier
http://dx.doi.org/10.1007/978-3-642-25361-4_6
Metadata
Show full item record
Author(s)
Esteban, Maria J. cc
Dolbeault, Jean cc
Abstract (EN)
This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).
Subjects / Keywords
optimal constants; compactness; existence; linearization; Emden-Fowler transformation; symmetry breaking; radial symmetry; extremal functions; logarithmic Sobolev inequality; logarithmic Hardy inequality; Gagliardo-Nirenberg inequality; Caffarelli-Kohn-Nirenberg inequality

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