Comptes Rendus
Mathematical analysis/Partial differential equations/Calculus of variations
Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities
Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 133-154.

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.

Nous utilisons le formalisme des entropies de Rényi pour établir le domaine de symétrie des fonctions extrémales dans une famille d'inégalités de Caffarelli–Kohn–Nirenberg sous-critiques. Par fonctions extrémales, il faut comprendre des fonctions qui réalisent le cas d'égalité dans les inégalités écrites avec des constantes optimales. La méthode étend des résultats récents sur les inégalités de Caffarelli–Kohn–Nirenberg critiques. En utilisant une heuristique donnée par une équation de diffusion non linéaire, nous donnons une preuve variationnelle d'un résultat de symétrie, grâce à un théorème de rigidité : dans la région de symétrie, tous les points critiques positifs sont à symétrie radiale et sont par conséquent égaux à l'unique point critique radial, positif, à une multiplication par une constante et à un changement d'échelle près. Ce résultat est optimal. La condition sur les paramètres est en effet complémentaire de celle qui définit la région dans laquelle il y a brisure de symétrie du fait de l'instabilité linéaire des fonctions radiales optimales. Comparé au cas critique, le domaine sous-critique nécessite de nouveaux outils. L'information de Fisher doit être remplacée par l'entropie de Rényi, et comme certaines invariances sont perdues, les estimations basées sur la transformation d'Emden–Fowler doivent être modifiées.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.01.004
Jean Dolbeault 1; Maria J. Esteban 1; Michael Loss 2; Matteo Muratori 3

1 Ceremade, UMR CNRS n° 7534, Université Paris-Dauphine, PSL Research University, place de Lattre-de-Tassigny, 75775 Paris cedex, France
2 School of Mathematics, Georgia Institute of Technology, Skiles Building, Atlanta GA 30332-0160, USA
3 Dipartimento di Matematica Felice Casorati, Università degli Studi di Pavia, Via A. Ferrata 5, 27100 Pavia, Italy
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Jean Dolbeault; Maria J. Esteban; Michael Loss; Matteo Muratori. Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 133-154. doi : 10.1016/j.crma.2017.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.004/

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