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Sharp Sobolev inequalities with lower order remainder terms


Authors: Olivier Druet, Emmanuel Hebey and Michel Vaugon
Journal: Trans. Amer. Math. Soc. 353 (2001), 269-289
MSC (2000): Primary 58E35
DOI: https://doi.org/10.1090/S0002-9947-00-02698-2
Published electronically: September 15, 2000
MathSciNet review: 1783789
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Abstract: Given a smooth compact Riemannian $n$-manifold $(M,g)$, this paper deals with the sharp Sobolev inequality corresponding to the embedding of $H_1^2(M)$ in $L^{2n/(n-2)}(M)$ where the $L^2$ remainder term is replaced by a lower order term.


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  • 1. Aubin, T., Problèmes isopérimétriques et espaces de Sobolev, Journal of Differential Geometry, 11, 1976, 573-598. MR 56:6711
  • 2. Aubin, T., Druet, O. and Hebey, E., Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature, Comptes Rendus de l'Académie des Sciences, Paris, Sér. I Math. 326, 1998, 1117-1121. MR 99j:53042
  • 3. Aubin, T. and Li, Y.Y., Sur la meilleure constante dans l'inégalité de Sobolev, Comptes Rendus de l'Académie des Sciences, Paris, Sér. I Math. 328, 1999, 135-138. MR 2000a:46048
  • 4. Beckner, W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Annals of Mathematics, 138, 1993, 213-242. MR 94m:58232
  • 5. Brezis, H. and Lieb, E.H., Sobolev inequalities with remainder terms, Journal of Functional Analysis, 62, 1985, 73-86. MR 86i:46033
  • 6. Brezis, H. and Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Mathematics, 36, 1983, 437-477. MR 84h:35059
  • 7. Caffarelli, L.A., Gidas, B. and Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Communications on Pure and Applied Mathematics, 42, 1989, 271-297. MR 90c:35075
  • 8. Croke, C.B., A sharp four dimensional isoperimetric inequality, Comment. Math. Helvetici, 59, 1984, 187-192. MR 85f:53060
  • 9. Djadli, Z. and Druet, O., Extremal functions and Sobolev inequalities on compact manifolds, Preprint of the university of Cergy-Pontoise, Volume 13, April 1999.
  • 10. Druet, O., Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, Journal of Functional Analysis, 159, 1998, 217-242. MR 99m:53076
  • 11. Druet, O., The best constants problem in Sobolev inequalities, Mathematische Annalen, 314, 1999, 327-346. MR 2000d:58033
  • 12. Druet, O., Hebey, E. and Vaugon, M., Optimal Nash's inequalities on Riemannian manifolds: the influence of geometry, International Mathematics Research Notices, 1995, 735-779. CMP 99:16
  • 13. Druet, O., Hebey, E. and Vaugon, M., Sharp Sobolev inequalities with lower order remainder terms, Preprint of the university of Cergy-Pontoise, Volume 3, January 1999.
  • 14. Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Second Edition, 1983. MR 86c:35035
  • 15. Hebey, E., Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, 1635, Springer-Verlag, 1996. MR 98k:46049
  • 16. Hebey, E., Fonctions extrémales pour une inégalité de Sobolev optimale dans la classe conforme de la sphère, Journal de Mathématiques Pures et Appliquées, 77, 1998, 721-733. MR 98j:58223
  • 17. Hebey, E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, Courant Institute of Mathematical Sciences, Volume 5, 1999. MR 2000e:58011
  • 18. Hebey, E., Humbert, E. and Vaugon, M., On the existence of extremal functions for the optimal $H_1^2$-Sobolev inequality after Djadli and Druet, Preprint, 1999.
  • 19. Hebey, E. and Vaugon, M., Meilleures constantes dans le théorème d'inclusion de Sobolev, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 13, 1996, 57-93. MR 96m:46054
  • 20. Hebey, E. and Vaugon, M., The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Mathematical Journal, 79, 1995, 235-279. MR 96c:53057
  • 21. Humbert, E., Best constants in the $L^2$-Nash inequality, Preprint, 1999.
  • 22. Kleiner, B., An isoperimetric comparison theorem, Inventiones Mathematicae, 108, 1992, 37-47. MR 92m:53056
  • 23. Obata, M., The conjectures on conformal transformations of Riemannian manifolds, Journal of Differential Geometry, 6, 1971, 247-258. MR 46:2601
  • 24. Talenti, G., Best constants in Sobolev inequality, Ann. di Matem. Pura ed Appl., 110, 1976, 353-372. MR 57:3846
  • 25. Trudinger, N.S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22, 1968, 265-274. MR 39:2093
  • 26. Weil, A., Sur les surfaces à courbure négative, Comptes Rendus de l'Académie des Sciences de Paris, 182, 1926, 1069-1071.

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Additional Information

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

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

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

DOI: https://doi.org/10.1090/S0002-9947-00-02698-2
Received by editor(s): June 15, 1999
Published electronically: September 15, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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