Inégalités de Sobolev optimales et inégalités isopérimétriques sur les variétés

Druet, Olivier

Séminaire de théorie spectrale et géométrie, Tome 20 (2001-2002), pp. 23-100.

[1] R.A. Adams, Sobolev Spaces, Academie Press, 1978. | Zbl

[2] W.X. Allard, On the first variation of a varifold, Annals of Mathematics 95 ( 1972), 417-491. | MR | Zbl

[3] C.B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannianpolyhedra, Transactions A.M.S. 53 ( 1943),101-129. | MR | Zbl

[4] F. Almgren, Existence and regularityalmost everywhere of solutions to elliptic variational problems with constraints, Mem. Am. Math. Soc. 165,4 ( 1976). | MR | Zbl

[5] F.V. Atkinson and Peletier L.A., Elliptic equations with nearly critical growth, J. Diff. Eq. 70 ( 1987), 349-365. | MR | Zbl

[6] T. Aubin, Equations différentiellesnon linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 ( 1976), 269-296. | MR | Zbl

[7] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 ( 1976), 573-598. | MR | Zbl

[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. | MR | Zbl

[9] T. Aubin and Y.Y. Li, On the best Sobolev inequality, J. Math. Pures Appl.. 78 ( 1999), 353-387. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] A. Beauville, Variétés Kàhleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 ( 1983), 755-782. | MR | Zbl

[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. | MR | Zbl

[14] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, 1971. | MR | Zbl

[15] A.L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10, Springer-Verlag, 1987. | MR | Zbl

[16] G. Bliss, An integral inequality, J. London Math. Soc. 5 ( 1930), 40-46. | JFM

[17] H. Brézis, Elliptic equations with limiting Sobolev exponents. The impact of topo logy, Comm. Pure Appl. Math. 39 ( 1986), 17-39. | MR | Zbl

[18] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 ( 1983), 437-477. | MR | Zbl

[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. | MR

[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). | Zbl

[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. | MR | Zbl

[22] M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992. | MR | Zbl

[23] S.Y.A. Chang and P.C. Yang, Compactness of isospectral conformal metrics in S3, Comment. Math. Helv. 64 ( 1989), 363-374. | MR | Zbl

[24] I. Chavel, Riemannian geometry: a modem introduction, Cambridge Tracts in Mathematics, Cambridge University Press, 1993. | MR | Zbl

[25] S. S. Chern. A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annais of Mathematics 45 ( 1944), 747-752. | MR | Zbl

[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). | MR | Zbl

[27] C.B. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 ( 1984), 187-192. | MR | Zbl

[28] Z. Djadli and O. Druet, Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var. 12 ( 2001), 59-84. | MR | Zbl

[29] O. Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 ( 1998), 217-242. | MR | Zbl

[30] O. Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 ( 1999), 327-346. | MR | Zbl

[31] O. Druet, Elliptic equations with critical Sobolev exponent in dimension 3, Ann. I.H.P., Analyse non-linéaire 19, 2 ( 2002), 125-142. | Numdam | MR | Zbl

[32] O. Druet, Inégalités de Sobolev-Poincaré en dimensions 4 et 5 et courbure scalaire négative ou nulle, preprint ( 2002).

[33] O. Druet, Isoperimetric inequalities on compact manifolds, Geometriae Dedicata 90 ( 2002), 217-236. | MR | Zbl

[34] O. Druet, Optimal Sobolev inequalities and extremal functions. The three-dimensional case, Indiana Univ. Math. J. 51, 1 ( 2002), 69-88. | MR | Zbl

[35] O. Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. A.M.S 130, 8 ( 2002), 2351-2361. | MR | Zbl

[36] O. Druet and E. Hebey, Extremal functions for sharp Sobolev inequalities, Prépublications de l'Université de Cergy-Pontoise ( 2000).

[37] O. Druet and E. Hebey, Asymptotics for sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Adv. Diff. Eq. 7, 12 ( 2002), 1409-1478. | MR | Zbl

[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). | Zbl

[39] O. Druet, E. Hebey, and E. Robert, Blow-up theory for elliptic PDEs in Riemannian geometry, preprint ( 2002). | Zbl

[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. | MR | Zbl

[41] O. Druet, E. Hebey, and M. Vaugon, Sharp Sobolev inequalities with lower order remainder terms, Transactions A.M.S. 353 ( 2001), 269-289. | MR | Zbl

[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] 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. | MR | Zbl

[44] L.C. Evans and R.E. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, 1992. | MR | Zbl

[45] Z. Faget, Best constants in Sobolev inequalities on Riemannian manifolds in the presence of symmetries, Potential Analysis 17 ( 2002), 105-124. | MR | Zbl

[46| H. Fédérer and W.H. Fleming, Normal and integral currents, Ann. of Math. 72 ( 1960), 458-520. | MR | Zbl

[47] W.H. Fleming and R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 ( 1960), 218-222. | MR | Zbl

[48] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerce Mat. 7 ( 1958), 102-137. | MR | Zbl

[49] S. Gallot. Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, S. M. F., Astérisque 163-164 ( 1988), 31-91. | Numdam | MR | Zbl

[50] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, 1993. | Zbl

[51] D. Gilbarg and N. Trüdinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin Heidelberg New York, 1977. | MR | Zbl

[52] M. Gromov, Paul Lévy's isoperimetric inequality, preprint I.H.E.S. ( 1980).

[53] M. Gromov, Partial Differential Relations, Ergebnisse der Mathemaiik und ihrer Grenzgebiete, vol. 9, Springer-Verlag, 1986. | MR | Zbl

[54] M. Gromov and H.B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Annals of Malhematics 111 ( 1980), 423-434. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[57] E. Hebey, Asymptotics for some quasilinear elliptic equations, J. Diff. and Int Eq. 9 ( 1996), 71-88. | MR | Zbl

[58] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag. 1996. | MR | Zbl

[59] E. Hebey, Introduction à l'analyse non-linéaire sur les variétés, Fondations, Diderot Editeurs, Arts et Sciences, 1997. | Zbl

[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. | MR | Zbl

[61] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, vol. 5, Courant Institute of Mathematical Sciences, 1999. | MR | Zbl

[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. | MR | Zbl

[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). | MR

[64] E. Hebey, Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Trans. A.M.S. 354 ( 2002), 1193-1213. | MR | Zbl

[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). | MR

[66] E. Hebey, Sharp Sobolev inequalities of second order, J. Geom. Analysis (à paraître). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[71] E. Hebey and M. Vaugon, Sobolev spaces in presence of symmetries, J. Math. Pures Appl. 76 ( 1997), 859-881. | MR | Zbl

[72] E. Hebey and M. Vaugon, Frombest constants to critical functions, Math. Z. 237 ( 2001), 737-767. | MR | Zbl

[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. | Numdam | MR | Zbl

[74] E. Humbert, Best constants in the L2-Nash inequality, Proc. Royal Soc. Edinburgh (à paraître). | Zbl

[75] S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 33 ( 1983), 151-165. | Numdam | MR | Zbl

[76] D. Johnson and E. Morgan, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana University Math. Journal 49,3 ( 2000), 1017-1041. | MR | Zbl

[77] B. Kleiner, An isoperimetric comparison theorem. Inven t. Math. 108 ( 1992), 37-47. | MR | Zbl

[78] M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 ( 1999), 347-353. | MR | Zbl

[79] Y.Y. Li, Prescribing scalar curvature on Sn and relaxed problems, part I, Journal of Differential Equations 120 ( 1995), 319-420. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[82] V.G. Maz'Ja, Sobolev Spaces, Springer-Verlag, 1985. | MR

[83] E. Morgan, Geometrie Measure Theory : a Beginner's Guide, Academie Press, 1995. | Zbl

[84] J. Nash, The embedding theorem for Riemannian manifolds, Annals of Mathematics 63 ( 1956), 20-63. | Zbl

[85] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 ( 1959), 116-162. | Numdam | MR | Zbl

[86] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 ( 1971), 247-258. | MR | Zbl

[87] R. Osserman, The isoperimetric inequality, Bull. A.M.S. 84 ( 1978), 1183-1238. | MR | Zbl

[88] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, NewYork, 1998. | MR | Zbl

[89] O. Rey, Proof of two conjectures of H. Brézis and L.A. Peletier, Manuscripta Mathematica 65 ( 1989), 19-37. | MR | Zbl

[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. | MR | Zbl

[91] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 ( 1984), 479-495. | MR | Zbl

[92] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Momecatini Notes ( 1987), 120-154. | MR | Zbl

[93] R. Schoen and S.T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 ( 1979), 159-183. | MR | Zbl

[94] R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere, Calculus of Variations and PDE's 4 ( 1996), 1-25. | MR | Zbl

[95] S.L. Sobolev, Sur un théorème d'analyse fonctionnelle, Mat. Sb. (N.S.) 46 ( 1938), 471-496. | JFM

[96] M. Struwe. A global compactness resuit for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 ( 1984), 511-517. | MR | Zbl

[97] M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer, 1996. | MR | Zbl

[98] G. Talenti, Best constants in Sobolev inequaliry, Ann. Math. Pura Appl. 110 ( 1976), 353-372. | MR | Zbl

[99] N.S. Trüdinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 ( 1968), 265-274. | Numdam | MR | Zbl

[100] A. Weil. Sur les surfaces à courbure négative, C.R. Acad. Sci. Paris 182 ( 1926), 1069-1071. | JFM

[101] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 ( 1960), 21-37. | MR | Zbl

[102] W.R. Ziemer, Weakly differentiable functions, Graduate Text in Mathematics, 120, Springer-Verlag, 1989. | MR | Zbl