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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Zoé Faget
Journal: Trans. Amer. Math. Soc. 360 (2008), 2303-2325
MSC (2000): Primary 46E35
Published electronically: December 11, 2007
MathSciNet review: 2373315
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Abstract: Let $ (M,g)$ be a Riemannian compact $ n$-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.


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Additional Information

Zoé Faget
Affiliation: Departement Mathematik, ETH-Zentrum, CH-8092, Zurich, Switzerland
Address at time of publication: Equipe Géométrie et Dynamique, Institut Mathématiques, 173 rue de Chevaleret, 75013 Paris, France
Email: zoe.faget@math.ethz.ch, fagetzoe@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04308-5
PII: S 0002-9947(07)04308-5
Keywords: Best constants, optimal Sobolev inequalities, exceptional case, concentration phenomenon
Received by editor(s): December 8, 2005
Published electronically: December 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.