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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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  • 1. Aubin, T., Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55, 269-296, 1976. MR 0431287 (55:4288)
  • 2. Beckner, W., On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc., 129, 1233-1246, 2001. MR 1709740 (2001g:35009)
  • 3. Caffarelli, L. A., Gidas, B., and Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42, 271-297, 1989. MR 0982351 (90c:35075)
  • 4. Collion, S., Fonctions critiques et équations aux dérivées partielles elliptiques sur les variétés riemanniennes compactes, Thèse de l'université Paris VI, 2004.
  • 5. Demengel, F., and Hebey, E., On some nonlinear equations involving the $ p$-Laplacian with critical Sobolev growth, Adv. Differential Equations, 3, 533-574, 1998. MR 1659246 (2000j:35095)
  • 6. Djadli, Z., and Druet, O., Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var. Partial Differential Equations, 12, 59-84, 2001. MR 1808107 (2002d:58042)
  • 7. Druet, O., Optimal Sobolev inequalities and extremal functions. The three-dimensional case, Indiana Univ. Math. J., 51, 69-88, 2002. MR 1896157 (2003b:58032)
  • 8. -, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 19, 125-142, 2002. MR 1902741 (2003f:35104)
  • 9. -, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc., 130, 2351-2361, 2002. MR 1897460 (2003b:53036)
  • 10. Druet, O., and Hebey, E., The $ AB$ program in geometric analysis. Sharp Sobolev inequalities and related problems, Memoirs of the American Mathematical Society, MEMO/160/761, 2002. MR 1938183 (2003m:58028)
  • 11. Druet, O., Hebey, E., and Robert, F., Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, Princeton University Press, vol. 45, Princeton, N.J., 2004. MR 2063399 (2005g:53058)
  • 12. Grushin, V.V., On a class of hypoelliptic operators, Math. Sbornik, 12, 458-475, 1970.
  • 13. Hebey, E., Asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth, Differential Integral Equations, 13, 1073-1080, 2000. MR 1775246 (2001g:35077)
  • 14. -, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, Courant Institute of Mathematical Sciences, Vol. 5, 1999. Second edition published by the American Mathematical Society, 2000. MR 1688256 (2000e:58011)
  • 15. Hebey, E., and Vaugon, M., The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79, 235-279, 1995. MR 1340298 (96c:53057)
  • 16. -, From best constants to critical functions, Math. Z., 237, 737-767, 2001. MR 1854089 (2002h:58061)
  • 17. Humbert, E., and Vaugon, M., The problem of prescribed critical functions, Ann. Global Anal. Geom., 28 (2005), 19-34. MR 2157345
  • 18. Obata, M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom., 6, 247-258, 1971/72. MR 0303464 (46:2601)
  • 19. Robert, F., Critical functions and optimal Sobolev inequalities, Math. Z., 249 (2005), 485-492. MR 2121735

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Additional Information

Stephane Collion
Affiliation: 150 bis rue Legendre, 75017 Paris, France

Emmanuel Hebey
Affiliation: Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

Michel Vaugon
Affiliation: Département de Mathématiques, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France

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.

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