Weighted interpolation inequalities: a perturbation approach

Comptes Rendus. Mathématique

Esteban

Loss

et al. 2017

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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 which 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 which 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. Symmétrie des fonctions extrémales pour des inégalités de Caffarelli-Kohn-Nirenberg sous-critiquesNous 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.

Section: A Family Of Subcritical Caffarelli-kohn-nirenberg Interpolatsupporting

confidence: 93%

“…To prove this result, we split the proof in several steps: we will first establish a uniform bound and a decay rate for w inspired by [17] in Lemmas 5.2, 5.3, and then follow the methodology of [14,Appendix] in the subsequent Lemma 5.4.…”

Section: Regularity and Decay Estimatesmentioning

confidence: 99%

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

Comptes Rendus. Mathématique

Esteban

Loss

et al. 2017

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“…The existence of a positive, finite constant λ R 1 can be deduced from elementary variational techniques as in [13]. We infer from the definition of v R that this inequality is equivalent to…”

Section: 4mentioning

confidence: 99%

Diffusion and kinetic transport with very weak confinement

Bouin

Kinetic &Amp; Related Models

Schmeiser³

2020

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“…The other endpoint is β = (d − 2) γ/d , in which case p = d /(d − 2): according to [11] (also see Section 5), either γ ≥ 0, symmetry holds and there exists a symmetric extremal function, or γ < 0, and then symmetry is broken but there is no extremal function. in all other cases, the existence of an extremal function for (14) follows from standard methods: see [11,16,32] for related results.…”

Section: Subcritical Caffarelli-kohn-nirenberg Inequalities With Thementioning

confidence: 99%

Symmetry and Symmetry Breaking: Rigidity and Flows in Elliptic Pdes

Proceedings of the International Congress of Mathematicians (ICM 2018)

Esteban

Loss

2019

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The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.