Martial Agueh; Shirin Boroushaki; Nassif Ghoussoub
A dual Moser–Onofri inequality and its extensions to higher dimensional spheres
Annales de la faculté des sciences de Toulouse Sér. 6, 26 no. 2 (2017), p. 217-233, doi: 10.5802/afst.1531
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Class. Math.: 39B62, 35J20, 34A34, 34A30
Keywords: Onofri inequality, duality, mass transport, prescribed Gaussian curvature, fast diffusion
Résumé - Abstract
We use optimal mass transport to provide a new proof and a dual formula to the Moser–Onofri inequality on $\mathbb{S}^2$. 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 $\mathbb{R}^2$ 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 $\mu _2(x)=\frac{1}{\pi ({1+|x|^2})^2}$ 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 $\mathbb{S}^n$ with $n\ge 2$. What is remarkable is that the corresponding free energy is again given by $F(\rho )=-n\rho ^{1-\frac{1}{n}}$, 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.
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