Sharp Sobolev inequalities with lower order remainder terms

Druet, Olivier; Hebey, Emmanuel; Vaugon, Michel

doi:10.1090/S0002-9947-00-02698-2

Sharp Sobolev inequalities with lower order remainder terms
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by Olivier Druet, Emmanuel Hebey and Michel Vaugon PDF

Trans. Amer. Math. Soc. 353 (2001), 269-289 Request permission

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.

References

Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
Thierry Aubin, Olivier Druet, and Emmanuel Hebey, Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9, 1117–1121 (English, with English and French summaries). MR 1647223, DOI 10.1016/S0764-4442(98)80072-4
Thierry Aubin and Yan Yan Li, Sur la meilleure constante dans l’inégalité de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 2, 135–138 (French, with English and French summaries). MR 1669011, DOI 10.1016/S0764-4442(99)80151-7
William Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242. MR 1230930, DOI 10.2307/2946638
Haïm Brezis and Elliott H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73–86. MR 790771, DOI 10.1016/0022-1236(85)90020-5
Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
Christopher B. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), no. 2, 187–192. MR 749103, DOI 10.1007/BF02566344
Djadli, Z. and Druet, O., Extremal functions and Sobolev inequalities on compact manifolds, Preprint of the university of Cergy-Pontoise, Volume 13, April 1999.
Olivier Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 (1998), no. 1, 217–242. MR 1654123, DOI 10.1006/jfan.1998.3264
Olivier Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327–346. MR 1697448, DOI 10.1007/s002080050297
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.
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.
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970, DOI 10.1007/BFb0092907
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256
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.
Emmanuel Hebey and Michel Vaugon, Meilleures constantes dans le théorème d’inclusion de Sobolev, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 1, 57–93 (French, with English and French summaries). MR 1373472, DOI 10.1016/S0294-1449(16)30097-X
Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279. MR 1340298, DOI 10.1215/S0012-7094-95-07906-X
Humbert, E., Best constants in the $L^2$-Nash inequality, Preprint, 1999.
Bruce Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), no. 1, 37–47. MR 1156385, DOI 10.1007/BF02100598
Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
S. Bergmann and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fund. Math. 33 (1939), 75–94 (French). MR 57, DOI 10.4064/fm-33-1-75-94
Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
Weil, A., Sur les surfaces à courbure négative, Comptes Rendus de l’Académie des Sciences de Paris, 182, 1926, 1069-1071.