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Comptes Rendus Mathématique
Volume 355, n° 2
pages 133-154 (février 2017)
Doi : 10.1016/j.crma.2017.01.004
Received : 20 May 2016 ;  accepted : 10 January 2017
Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities
Symétrie des fonctions extrémales pour des inégalités de Caffarelli–Kohn–Nirenberg sous-critiques

Fig. 1

Fig. 1 : 

In dimension d =4, with p =1.2, the grey area corresponds to the cone determined by d 2+(γ d )/p β <(d 2)γ /d and γ (−∞,d ) in (2). The light grey area is the region of symmetry, while the dark grey area is the region of symmetry breaking. The threshold is determined by the hyperbola (d γ )2(β d +2)24(d 1)=0 or, equivalently β =β FS (γ ). Notice that the condition p p induces the restriction β d 2+(γ d )/p , so that the region of symmetry is bounded. The largest possible cone is achieved as p 1 and is limited from below by the condition β >γ 2.

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