Sharp Sobolev inequalities in the presence of a twist
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- by Stephane Collion, Emmanuel Hebey and Michel Vaugon PDF
- Trans. Amer. Math. Soc. 359 (2007), 2531-2537 Request permission
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 $u \in H_1^2(M)$, \[ \left (\int _M\vert u\vert ^{2^\star }dv_g\right )^{2/2^\star }\le K \int _MA_x(\nabla u, \nabla u)dv_g + B\int _Mu^2dv_g \] 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.References
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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
- Received by editor(s): January 28, 2005
- Published electronically: January 4, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2531-2537
- MSC (2000): Primary 58E35
- DOI: https://doi.org/10.1090/S0002-9947-07-03959-1
- MathSciNet review: 2286043