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Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Authors:	Zoé Faget

Institution:	Departement Mathematik, ETH-Zentrum, CH-8092, Zurich, Switzerland

Abstract:	Let be a Riemannian compact -manifold. We know that for any $\varepsilon>0$ , there exists $C_\varepsilon>0$ such that for any $u\in H_1^n(M)$ , $\int_Me^u\mathrm{dv}_g\le C_\varepsilon\exp(\mu_n+\varepsilon)\int_M\vert\nabla u\vert^n\mathrm{dv}_g+\frac{1}{\mathrm{vol}(M)}\int_Mu\mathrm{dv}_g]$ , $\mu_n$ being the smallest constant possible such that the inequality remains true for any $u\in H_1^n(M)$ . We call $\mu_n$ the ``first best constant'. We prove in this paper that it is possible to choose $\varepsilon=0$ and keep $C_\varepsilon$ a finite constant. In other words we prove the existence of a ``second best constant' in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.

Keywords:	Best constants optimal Sobolev inequalities exceptional case concentration phenomenon

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