A dual Moser–Onofri inequality and its extensions to higher dimensional spheres

Agueh, Martial; Boroushaki, Shirin; Ghoussoub, Nassif

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 217-233.

Nous utilisons une méthode de transport optimal pour donner une nouvelle démonstration et une forme duale de l’inégalité de Moser-Onofri sur ${𝕊}^2$. Cette approche est dans le même esprit que celle des inégalités de Sobolev et de Gagliardi-Nirenberg par Cordero-Erausquin, Nazaret et Villani [5] ainsi que de leurs généralisations par Agueh–Ghoussoub–Kang [1]. Il y a néanmoins plusieurs difficultés nouvelles qui apparaissent une fois qu’on a effectué une projection stéréographique sur ${ℝ}^2$ : les fonctions n’ont plus support compact, ce qui demande de tenir compte de termes de bord. De plus, l’énergie libre duale de la densité probabilité de référence ${\mu }_2\left(x\right)=\frac{1}{{\pi (1+|x|}^2{)}^2}$ n’est pas finie sur l’espace entier ce qui demande d’introduire une énergie libre renormalisée dans la formule duale. Nous étendons aussi cette inégalité en dimensions supérieures et établissons une inégalité d’Onofri pour les sphères ${𝕊}^n$ quand $n\ge 2$. Il est remarquable que l’énergie libre correspondante est toujours donnée par $F\left(\rho \right)=-n{\rho }^{1-\frac{1}{n}}$, ce qui signifie que les problèmes de courbure scalaire prescrite et de courbure de Gauss prescrite conduisent essentiellement au même problème dual.

We use optimal mass transport to provide a new proof and a dual formula to the Moser–Onofri inequality on ${𝕊}^2$. This is in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani [5] to the Sobolev and Gagliardo–Nirenberg inequalities and the one of Agueh–Ghoussoub–Kang [1] to more general settings. There are however many hurdles to overcome once a stereographic projection on ${ℝ}^2$ is performed: Functions are not necessarily of compact support, hence boundary terms need to be evaluated. Moreover, the corresponding dual free energy of the reference probability density ${\mu }_2\left(x\right)=\frac{1}{\pi {\left(1+{\left|x\right|}^2\right)}^2}$ is not finite on the whole space, which requires the introduction of a renormalized free energy into the dual formula. We also extend this duality to higher dimensions and establish an extension of the Onofri inequality to spheres ${𝕊}^n$ with $n\ge 2$. What is remarkable is that the corresponding free energy is again given by $F\left(\rho \right)=-n{\rho }^{1-\frac{1}{n}}$, which means that both the prescribed scalar curvature problem and the prescribed Gaussian curvature problem lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.

Classification : 39B62, 35J20, 34A34, 34A30
Mots clés : Onofri inequality, duality, mass transport, prescribed Gaussian curvature, fast diffusion

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