Inégalités de Sobolev optimales et inégalités isopérimétriques sur les variétés

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1. [1] R.A. Adams, Sobolev Spaces, Academie Press, 1978. Zbl1098.46001
2. [2] W.X. Allard, On the first variation of a varifold, Annals of Mathematics 95 ( 1972), 417-491. Zbl0252.49028MR307015
3. [3] C.B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannianpolyhedra, Transactions A.M.S. 53 ( 1943),101-129. Zbl0060.38102MR7627
4. [4] F. Almgren, Existence and regularityalmost everywhere of solutions to elliptic variational problems with constraints, Mem. Am. Math. Soc. 165,4 ( 1976). Zbl0327.49043MR420406
5. [5] F.V. Atkinson and Peletier L.A., Elliptic equations with nearly critical growth, J. Diff. Eq. 70 ( 1987), 349-365. Zbl0657.35058MR915493
6. [6] T. Aubin, Equations différentiellesnon linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 ( 1976), 269-296. Zbl0336.53033MR431287
7. [7] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 ( 1976), 573-598. Zbl0371.46011MR448404
8. [8] T. Aubin, O. Druet, and E. Hebey, Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature, C.R. Acad. Sci. Paris Sér. I Math. 326 ( 1998), 1117-1121. Zbl0908.53021MR1647223
9. [9] T. Aubin and Y.Y. Li, On the best Sobolev inequality, J. Math. Pures Appl.. 78 ( 1999), 353-387. Zbl0944.46027MR1696357
10. [10] D. Bakry. L'hypercomractivité et son utilisation en théorie des semi-groupes, Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Mathematics, vol. 1581, Springer, Berlin, 1994, pp. 1-114. Zbl0856.47026MR1307413
11. [11] D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J. 85 ( 1996), 253-270. Zbl0870.60071MR1412446
12. [12] A. Beauville, Variétés Kàhleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 ( 1983), 755-782. Zbl0537.53056MR730926
13. [13] H. Beresrycki, L. Nirenberg, and S. Varadhan, The principal eigenvalue and maximum principlefor second order elliptic operators in general domains, Comm. Pure Appl. Math. 47 ( 1994), 47-92. Zbl0806.35129MR1258192
14. [14] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, 1971. Zbl0223.53034MR282313
15. [15] A.L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10, Springer-Verlag, 1987. Zbl0613.53001MR867684
16. [16] G. Bliss, An integral inequality, J. London Math. Soc. 5 ( 1930), 40-46. Zbl56.0434.02JFM56.0434.02
17. [17] H. Brézis, Elliptic equations with limiting Sobolev exponents. The impact of topo logy, Comm. Pure Appl. Math. 39 ( 1986), 17-39. Zbl0601.35043MR861481
18. [18] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 ( 1983), 437-477. Zbl0541.35029MR709644
19. [19] H. Brézis and L.A. Peletier, Asymptotics for elliptic equations in volving critical Sobolev exponents, Partial Differential Equations and the Calculus of Variations (L Modica F. Colombini, A. Marino and S. Spagnolo, eds.), Basel, Birkhaüser, 1989. MR1034005
20. [20] C. Brouttelande, The best constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold, Proc. Roy. Soc. Edinburgh (à paraître). Zbl1031.58009
21. [21] L.A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 ( 1989), 271-297. Zbl0702.35085MR982351
22. [22] M. do Carmo, Riemannian Geometry, Birkhäuser, 1992. Zbl0752.53001MR1138207
23. [23] S.Y.A. Chang and P.C. Yang, Compactness of isospectral conformal metrics in S3, Comment. Math. Helv. 64 ( 1989), 363-374. Zbl0679.53038MR998854
24. [24] I. Chavel, Riemannian geometry: a modem introduction, Cambridge Tracts in Mathematics, Cambridge University Press, 1993. Zbl0810.53001MR2229062
25. [25] S. S. Chern. A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annais of Mathematics 45 ( 1944), 747-752. Zbl0060.38103MR11027
26. [26] D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, preprint, disponible sur http://www.umpa.ens-lyon.fr/ bnazaret ( 2002). Zbl1048.26010MR2032031
27. [27] C.B. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 ( 1984), 187-192. Zbl0552.53017MR749103
28. [28] Z. Djadli and O. Druet, Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var. 12 ( 2001), 59-84. Zbl0998.58008MR1808107
29. [29] O. Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 ( 1998), 217-242. Zbl0923.46035MR1654123
30. [30] O. Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 ( 1999), 327-346. Zbl0934.53028MR1697448
31. [31] O. Druet, Elliptic equations with critical Sobolev exponent in dimension 3, Ann. I.H.P., Analyse non-linéaire 19, 2 ( 2002), 125-142. Zbl1011.35060MR1902741
32. [32] O. Druet, Inégalités de Sobolev-Poincaré en dimensions 4 et 5 et courbure scalaire négative ou nulle, preprint ( 2002). 
33. [33] O. Druet, Isoperimetric inequalities on compact manifolds, Geometriae Dedicata 90 ( 2002), 217-236. Zbl1025.58014MR1898162
34. [34] O. Druet, Optimal Sobolev inequalities and extremal functions. The three-dimensional case, Indiana Univ. Math. J. 51, 1 ( 2002), 69-88. Zbl1037.58012MR1896157
35. [35] O. Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. A.M.S 130, 8 ( 2002), 2351-2361. Zbl1067.53026MR1897460
36. [36] O. Druet and E. Hebey, Extremal functions for sharp Sobolev inequalities, Prépublications de l'Université de Cergy-Pontoise ( 2000). 
37. [37] O. Druet and E. Hebey, Asymptotics for sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Adv. Diff. Eq. 7, 12 ( 2002), 1409-1478. Zbl1044.58031MR1920541
38. [38] O. Druet and E. Hebey. The AB program in geometrie analysis. Sharp Sobolev inequalities and relaled problems, Memoirs of the A.M.S. (à paraître). Zbl1023.58009
39. [39] O. Druet, E. Hebey, and E. Robert, Blow-up theory for elliptic PDEs in Riemannian geometry, preprint ( 2002). Zbl1059.58017
40. [40] O. Druet, E. Hebey, and M. Vaugon, Optimal Nash's inequalities on Riemannian manifolds: the influence of geometry. I.M.R.N. 14 ( 1999), 735-779. Zbl0959.58043MR1704184
41. [41] O. Druet, E. Hebey, and M. Vaugon, Sharp Sobolev inequalities with lower order remainder terms, Transactions A.M.S. 353 ( 2001), 269-289. Zbl0968.58012MR1783789
42. [42] O. Druet and F. Robert, Asymptotic profile and blow-up estimates on compact Riemannian manifolds, Prépublications de l'Université de Cergy-Pontoise ( 1999). 
43. [43] O. Druet and F. Robert, Asymptotic profile for the sub-extremals of the sharp Sobolev inequality on the sphere, Comm. P.D.E. 26 (5-6) ( 2001), 743-778. Zbl0998.58010MR1843283
44. [44] L.C. Evans and R.E. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, 1992. Zbl0804.28001MR1158660
45. [45] Z. Faget, Best constants in Sobolev inequalities on Riemannian manifolds in the presence of symmetries, Potential Analysis 17 ( 2002), 105-124. Zbl1044.58032MR1908673
46. [46| H. Fédérer and W.H. Fleming, Normal and integral currents, Ann. of Math. 72 ( 1960), 458-520. Zbl0187.31301MR123260
47. [47] W.H. Fleming and R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 ( 1960), 218-222. Zbl0094.26301MR114892
48. [48] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerce Mat. 7 ( 1958), 102-137. Zbl0089.09401MR102740
49. [49] S. Gallot. Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, S. M. F., Astérisque 163-164 ( 1988), 31-91. Zbl0674.53001MR999971
50. [50] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, 1993. Zbl0636.53001
51. [51] D. Gilbarg and N. Trüdinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin Heidelberg New York, 1977. Zbl0361.35003MR473443
52. [52] M. Gromov, Paul Lévy's isoperimetric inequality, preprint I.H.E.S. ( 1980). 
53. [53] M. Gromov, Partial Differential Relations, Ergebnisse der Mathemaiik und ihrer Grenzgebiete, vol. 9, Springer-Verlag, 1986. Zbl0651.53001MR864505
54. [54] M. Gromov and H.B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Annals of Malhematics 111 ( 1980), 423-434. Zbl0463.53025MR577131
55. [55] M. Gromov and H.B. Lawson, Spin and scalar curvature in the presence of a fundamental group, Annals of Mathematics 111 ( 1980), 209-230. Zbl0445.53025MR569070
56. [56] Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolez exponent, Ann. I.H.R, Analyse non-linéaire 8,2 ( 1991), 159-174. Zbl0729.35014MR1096602
57. [57] E. Hebey, Asymptotics for some quasilinear elliptic equations, J. Diff. and Int Eq. 9 ( 1996), 71-88. Zbl0842.35034MR1364035
58. [58] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag. 1996. Zbl0866.58068MR1481970
59. [59] E. Hebey, Introduction à l'analyse non-linéaire sur les variétés, Fondations, Diderot Editeurs, Arts et Sciences, 1997. Zbl0918.58001
60. [60] E. Hebey, Fonctions extrémales pour une inégalitéde Sobolev optimale dans la classe conforme de lasphère, J. Math. Pures et Appl. 77 ( 1998), 721-733. Zbl0914.53026MR1645089
61. [61] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, vol. 5, Courant Institute of Mathematical Sciences, 1999. Zbl0981.58006MR1688256
62. [62] E. Hebey, Asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth. J .Diff. and Int. Eq. 13 ( 2000), 1073-1080. Zbl0976.35022MR1775246
63. [63] E. Hebey, Nonlinear elliptic equations of critical Sobolev growth from a dynamical viewpoint, Preprint, disponible sur http://www.u-cergy.fr/rech/pages/hebey/index.html ( 2002). Zbl1210.35092MR2082394
64. [64] E. Hebey, Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Trans. A.M.S. 354 ( 2002), 1193-1213. Zbl0992.58017MR1867378
65. [65] E. Hebey, Sharp Sobolev-Poincaré inequalities on compaet Riemannian manifolds. notes from various lectures, Preprint, disponible sur http://www.u-cergy.fr/rech/pages/hebey/index.html ( 2002). MR1867378
66. [66] E. Hebey, Sharp Sobolev inequalities of second order, J. Geom. Analysis (à paraître). Zbl1032.58008MR1967041
67. [67] E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz operators with constant coefficients, Calc. Var. PDE's 13 ( 2001), 491-517. Zbl0998.58007MR1867939
68. [68] E. Hebey and M. Vaugon, Meilleures constantes dans le théorème d'inclusion de Sobolev et multiplicité pour les problèmes de Nirenberget Yamabe, Indiana Univ. Math. J. 41 ( 1992), 377-407. Zbl0764.53029MR1183349
69. [69] E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 ( 1995), 235-279. Zbl0839.53030MR1340298
70. [70] E. Hebey and M. Vaugon, Meilleures constames dans le théorème d'inclusion de Sobolev, Ann. Inst. H. Poincare. Anal. Non Linéaire 13 ( 1996), 57-93. Zbl0849.53035MR1373472
71. [71] E. Hebey and M. Vaugon, Sobolev spaces in presence of symmetries, J. Math. Pures Appl. 76 ( 1997), 859-881. Zbl0886.58003MR1489942
72. [72] E. Hebey and M. Vaugon, Frombest constants to critical functions, Math. Z. 237 ( 2001), 737-767. Zbl0992.58016MR1854089
73. [73] E. Heinize and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Scient. Ec. Norm. Sup. 11 ( 1978), 451-470. Zbl0416.53027MR533065
74. [74] E. Humbert, Best constants in the L2-Nash inequality, Proc. Royal Soc. Edinburgh (à paraître). Zbl1040.53048
75. [75] S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 33 ( 1983), 151-165. Zbl0528.53040MR699492
76. [76] D. Johnson and E. Morgan, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana University Math. Journal 49,3 ( 2000), 1017-1041. Zbl1021.53020MR1803220
77. [77] B. Kleiner, An isoperimetric comparison theorem. Inven t. Math. 108 ( 1992), 37-47. Zbl0770.53031MR1156385
78. [78] M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 ( 1999), 347-353. Zbl0953.53025MR1685586
79. [79] Y.Y. Li, Prescribing scalar curvature on Sn and relaxed problems, part I, Journal of Differential Equations 120 ( 1995), 319-420. Zbl0827.53039MR1347349
80. [80] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Parti Ann. Inst. H. Poincaré 1 ( 1984), 109-145. Zbl0541.49009MR778970
81. [81] P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part I, Rev. Mat. Iberoamericano 1.1 ( 1985), 145-201. Zbl0704.49005MR834360
82. [82] V.G. Maz'ja, Sobolev Spaces, Springer-Verlag, 1985. MR817985
83. [83] E. Morgan, Geometrie Measure Theory : a Beginner's Guide, Academie Press, 1995. Zbl0819.49024
84. [84] J. Nash, The embedding theorem for Riemannian manifolds, Annals of Mathematics 63 ( 1956), 20-63. Zbl0070.38603
85. [85] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 ( 1959), 116-162. Zbl0088.07601MR109940
86. [86] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 ( 1971), 247-258. Zbl0236.53042MR303464
87. [87] R. Osserman, The isoperimetric inequality, Bull. A.M.S. 84 ( 1978), 1183-1238. Zbl0411.52006MR500557
88. [88] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, NewYork, 1998. Zbl0914.53001MR1480173
89. [89] O. Rey, Proof of two conjectures of H. Brézis and L.A. Peletier, Manuscripta Mathematica 65 ( 1989), 19-37. Zbl0708.35032MR1006624
90. [90] F. Robert, Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent : the radial case. Advances in Diff. Eq. 6,7 ( 2001), 821-846. Zbl1087.35026MR1828998
91. [91] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 ( 1984), 479-495. Zbl0576.53028MR788292
92. [92] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Momecatini Notes ( 1987), 120-154. Zbl0702.49038MR994021
93. [93] R. Schoen and S.T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 ( 1979), 159-183. Zbl0423.53032MR535700
94. [94] R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere, Calculus of Variations and PDE's 4 ( 1996), 1-25. Zbl0843.53037MR1379191
95. [95] S.L. Sobolev, Sur un théorème d'analyse fonctionnelle, Mat. Sb. (N.S.) 46 ( 1938), 471-496. Zbl64.1100.02JFM64.1100.02
96. [96] M. Struwe. A global compactness resuit for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 ( 1984), 511-517. Zbl0535.35025MR760051
97. [97] M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer, 1996. Zbl0864.49001MR1411681
98. [98] G. Talenti, Best constants in Sobolev inequaliry, Ann. Math. Pura Appl. 110 ( 1976), 353-372. Zbl0353.46018MR463908
99. [99] N.S. Trüdinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 ( 1968), 265-274. Zbl0159.23801MR240748
100. [100] A. Weil. Sur les surfaces à courbure négative, C.R. Acad. Sci. Paris 182 ( 1926), 1069-1071. Zbl52.0712.05JFM52.0712.05
101. [101] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 ( 1960), 21-37. Zbl0096.37201MR125546
102. [102] W.R. Ziemer, Weakly differentiable functions, Graduate Text in Mathematics, 120, Springer-Verlag, 1989. Zbl0692.46022MR1014685