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

 

Sharp Sobolev inequalities in the presence of a twist


Authors: Stephane Collion, Emmanuel Hebey and Michel Vaugon
Journal: Trans. Amer. Math. Soc. 359 (2007), 2531-2537
MSC (2000): Primary 58E35
Published electronically: January 4, 2007
MathSciNet review: 2286043
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Abstract: Let $ (M,g)$ be a smooth compact Riemannian manifold of dimension $ n \ge 3$. Let also $ A$ be a smooth symmetrical positive $ (0,2)$-tensor field in $ M$. By the Sobolev embedding theorem, we can write that there exist $ K,B>0$ such that for any $ u \in H_1^2(M)$,

$\displaystyle \left(\int_M\vert u\vert^{2^\star}dv_g\right)^{2/2^\star}\le K \int_MA_x(\nabla u, \nabla u)dv_g + B\int_Mu^2dv_g $

where $ H_1^2(M)$ is the standard Sobolev space of functions in $ L^2$ with one derivative in $ L^2$. We investigate in this paper the value of the sharp $ K$ in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.


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Additional Information

Stephane Collion
Affiliation: 150 bis rue Legendre, 75017 Paris, France
Email: Stephane.Collion@wanadoo.fr

Emmanuel Hebey
Affiliation: Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Email: Emmanuel.Hebey@math.u-cergy.fr

Michel Vaugon
Affiliation: Département de Mathématiques, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France
Email: vaugon@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-07-03959-1
PII: S 0002-9947(07)03959-1
Received by editor(s): January 28, 2005
Published electronically: January 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.