A dual Moser-Onofri inequality and its extensions to higher dimensional spheres
| Autor: | Agueh, Martial, Boroushaki, Shirin, Ghoussoub, Nassif |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on $\s^2$ in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani to the Sobolev inequality and of Agueh-Ghoussoub-Kang 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 $\mu_2(x)=\frac{1}{\pi(1+|x|^2)^2}$ 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 $\s^n$ with $n\geq 2$. 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 {\it prescribed scalar curvature problem} and the {\it prescribed Gaussian curvature problem} lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations. Comment: 15 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ |
| Databáze: | arXiv |
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