Browse issues
no. 1
Sobolev’s inequality under a curvature-dimension condition
Louis Dupaigne1; Ivan Gentil1; Simon Zugmeyer2
1 Institut Camille Jordan, UMR CNRS 5208, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex
2 S. Z. MAP5, UMR CNRS 8154, Université de Paris, 45 rue des Saints-Pères, 75270 Paris cedex 06
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 1, pp. 125-144.

In this note we present a new proof of Sobolev’s inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut–Véron and Véron’s 1991 classical work.

Dans cette note, nous proposons une nouvelle preuve de l’inégalité de Sobolev sur les variétés à courbure de Ricci minorée par une constante positive. Le résultat avait été obtenu en 1983 par Ilias. Nous présentons une preuve très courte de ce théorème, dressons l’état de l’art pour cette fameuse inégalité et expliquons en quoi notre méthode, qui repose sur un flot de gradient, est simple et robuste. En particulier, nous élucidons les calculs utilisés dans des travaux précédents, à commencer par un célèbre article de Bidaut–Véron et Véron publié en 1991.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1731

Louis Dupaigne 1; Ivan Gentil 1; Simon Zugmeyer 2

1 Institut Camille Jordan, UMR CNRS 5208, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex
2 S. Z. MAP5, UMR CNRS 8154, Université de Paris, 45 rue des Saints-Pères, 75270 Paris cedex 06
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2023_6_32_1_125_0,
     author = {Louis Dupaigne and Ivan Gentil and Simon Zugmeyer},
     title = {Sobolev{\textquoteright}s inequality under a curvature-dimension condition},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {125--144},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 32},
     number = {1},
     year = {2023},
     doi = {10.5802/afst.1731},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1731/}
}
TY  - JOUR
AU  - Louis Dupaigne
AU  - Ivan Gentil
AU  - Simon Zugmeyer
TI  - Sobolev’s inequality under a curvature-dimension condition
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2023
SP  - 125
EP  - 144
VL  - 32
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1731/
DO  - 10.5802/afst.1731
LA  - en
ID  - AFST_2023_6_32_1_125_0
ER  - 
%0 Journal Article
%A Louis Dupaigne
%A Ivan Gentil
%A Simon Zugmeyer
%T Sobolev’s inequality under a curvature-dimension condition
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2023
%P 125-144
%V 32
%N 1
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1731/
%R 10.5802/afst.1731
%G en
%F AFST_2023_6_32_1_125_0
Louis Dupaigne; Ivan Gentil; Simon Zugmeyer. Sobolev’s inequality under a curvature-dimension condition. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 1, pp. 125-144. doi : 10.5802/afst.1731. https://afst.centre-mersenne.org/articles/10.5802/afst.1731/

[1] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Gradient flows in metric spaces and in the space of probability measures., Birkhäuser, 2008, vii+334 pages | Zbl

[2] Thierry Aubin Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., Volume 55 (1976), pp. 269-296 | Zbl

[3] Thierry Aubin Problèmes isoperimetriques et espaces de Sobolev, J. Differ. Geom., Volume 11 (1976), pp. 573-598 | Zbl

[4] Dominique Bakry L’hypercontractivité et son utilisation en théorie des semigroupes., Lectures on probability theory. Saint-Flour 1992, Springer, 1994, pp. 1-114 | Zbl

[5] Dominique Bakry; Ivan Gentil; Michel Ledoux Analysis and geometry of Markov diffusion operators., Springer, 2014, xx+552 pages | DOI

[6] Dominique Bakry; Michel Ledoux Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J., Volume 85 (1996) no. 1, pp. 253-270 | MR | Zbl

[7] William Beckner Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math., Volume 138 (1993) no. 1, pp. 213-242 | DOI | MR | Zbl

[8] Marcel Berger; Paul Gauduchon; Edmond Mazet Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, 194, Springer, 1971 | DOI | Zbl

[9] Marie-Françoise Bidaut-Veron; Laurent Véron Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., Volume 106 (1991) no. 3, pp. 489-539 | DOI | MR | Zbl

[10] Haim Brezis Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, 2011, xiii+599 pages | DOI | Zbl

[11] Luis A. Caffarelli; Basilis Gidas; Joel Spruck Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 271-297 | DOI | MR | Zbl

[12] José A. Carrillo; Ansgar Jüngel; Peter A. Markowich; Giuseppe Toscani; Andreas Unterreiter Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001) no. 1, pp. 1-82 | DOI | MR | Zbl

[13] José A. Carrillo; Giuseppe Toscani Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., Volume 49 (2000) no. 1, pp. 113-142 | MR | Zbl

[14] José A. Carrillo; Juan Luis Vázquez Fine asymptotics for fast diffusion equations, Commun. Partial Differ. Equations, Volume 28 (2003) no. 5-6, pp. 1023-1056 | DOI | MR | Zbl

[15] Fabio Cavalletti; Andrea Mondino Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds, Geom. Topol., Volume 21 (2017) no. 1, pp. 603-645 | DOI | MR | Zbl

[16] Dario Cordero-Erausquin; Bruno Nazaret; Cédric Villani A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities, Adv. Math., Volume 182 (2004) no. 2, pp. 307-332 | DOI | MR | Zbl

[17] Manuel Del Pino; Jean Dolbeault Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., Volume 81 (2002) no. 9, pp. 847-875 | DOI | MR | Zbl

[18] Jérôme Demange Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie, Ph. D. Thesis, Université Paul Sabatier Toulouse 3 (France) (2005)

[19] Jérôme Demange Improved Gagliardo–Nirenberg–Sobolev inequalities on manifolds with positive curvature., J. Funct. Anal., Volume 254 (2008) no. 3, pp. 593-611 | DOI | MR | Zbl

[20] Jean Dolbeault; Maria J. Esteban; Michal Kowalczyk; Michael Loss Sharp interpolation inequalities on the sphere: new methods and consequences, Partial differential equations. Theory, control and approximation, Springer, 2014, pp. 225-242 | DOI | Zbl

[21] Jean Dolbeault; Maria J. Esteban; Michael Loss Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., Volume 267 (2014) no. 5, pp. 1338-1363 | DOI | MR | Zbl

[22] Jean Dolbeault; Maria J. Esteban; Michael Loss Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., Volume 206 (2016) no. 2, pp. 397-440 | DOI | MR | Zbl

[23] Éric Fontenas Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères, Bull. Sci. Math., Volume 121 (1997) no. 2, pp. 71-96 | MR | Zbl

[24] Emilio Gagliardo Proprietà di alcune classi di funzioni in più variabili, Ric. Mat., Volume 7 (1958), pp. 102-137 | Zbl

[25] Ivan Gentil L’entropie, de Clausius aux inégalités fonctionnelles (2020) (https://hal.archives-ouvertes.fr/hal-02464182)

[26] Ivan Gentil; Christian Léonard; Luigia Ripani Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view, Rev. Mat. Iberoam., Volume 36 (2020) no. 4, pp. 1071-1112 | DOI | MR | Zbl

[27] Basilis Gidas; Wei-Ming Ni; Louis Nirenberg Symmetry of positive solutions of nonlinear elliptic equations in ℝ n , Volume 7A, 1981, pp. 369-402 | Zbl

[28] Basilis Gidas; Joel Spruck Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., Volume 34 (1981), pp. 525-598 | DOI | MR | Zbl

[29] Nicola Gigli Second order analysis on (𝒫 2 (M),W 2 )., Mem. Am. Math. Soc., Volume 1018 (2012), p. 154 | DOI | Zbl

[30] Misha Gromov Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser, 2007, xx+585 pages (transl. from the French by Sean Michael Bates.) | Zbl

[31] Emmanuel Hebey Introduction à l’analyse non linéaire sur les variétés, Diderot Editeur, 1997, viii+406 pages | Zbl

[32] Emmanuel Hebey Nonlinear analysis on manifolds: Sobolev spaces and inequalities, 5, American Mathematical Society, 2000, xii+290 pages | Zbl

[33] Said Ilias Constantes explicites pour les inégalités de Sobolev sur les variétés Riemanniennes compactes, Ann. Inst. Fourier, Volume 33 (1983) no. 2, pp. 151-165 | DOI | Numdam | MR | Zbl

[34] Said Ilias; Abdolhakim Shouman Sobolev inequalities on a weighted Riemannian manifold of positive Bakry-Émery curvature and convex boundary, Pac. J. Math., Volume 294 (2018) no. 2, pp. 423-451 | DOI | Zbl

[35] Richard Jordan; David Kinderlehrer; Felix Otto The variational formulation of the Fokker-Planck equation., SIAM J. Math. Anal., Volume 29 (1998) no. 1, pp. 1-17 | DOI | MR | Zbl

[36] Olga A. Ladyženskaja; Vsevolod A. Solonnikov; Nina N. Uralceva Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968, xi+648 pages (Translated from the Russian by S. Smith) | DOI | MR

[37] John M. Lee; Thomas H. Parker The Yamabe problem, Bull. Am. Math. Soc., Volume 17 (1987), pp. 37-91 | MR | Zbl

[38] Jean René Licois; Laurent Véron Un théorème d’annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Math. Acad. Sci. Paris, Volume 320 (1995) no. 11, pp. 1337-1342 | Zbl

[39] Jean René Licois; Laurent Véron A class of nonlinear conservative elliptic equations in cylinders, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 26 (1998) no. 2, pp. 249-283 | Numdam | MR | Zbl

[40] Elliott H. Lieb Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math., Volume 118 (1983), pp. 349-374 | DOI | MR | Zbl

[41] Louis Nirenberg On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser., Volume 13 (1959), pp. 115-162 | Numdam | MR | Zbl

[42] Ivan Y. Nobili Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds, Calc. Var. Partial Differ. Equ., Volume 61 (2022), 180 | DOI | MR | Zbl

[43] Morio Obata Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, Volume 14 (1962), pp. 333-340 | MR | Zbl

[44] Felix Otto The geometry of dissipative evolution equations: The porous medium equation., Commun. Partial Differ. Equations, Volume 26 (2001) no. 1-2, pp. 101-174 | DOI | MR | Zbl

[45] Felix Otto; Cédric Villani Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality., J. Funct. Anal., Volume 173 (2000) no. 2, pp. 361-400 | DOI | MR | Zbl

[46] Angelo Profeta The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds, Potential Anal., Volume 43 (2015) no. 3, pp. 513-529 | DOI | MR | Zbl

[47] Eugene Rodemich The Sobolev inequalities with best possible constants (1966) (Analysis seminar at California Institute of Technology)

[48] Oscar S. Rothaus Hypercontractivity and the Bakry-Emery criterion for compact Lie groups, J. Funct. Anal., Volume 65 (1986), pp. 358-367 | DOI | MR | Zbl

[49] Sergei Sobolev Sur un théorème d’analyse fonctionnelle, Rec. Math. Moscou, n. Ser., Volume 4 (1938), pp. 471-497 | Zbl

[50] Guido Stampacchia Equations elliptiques du second ordre à coefficients discontinus, Séminaire de mathématiques supérieures (été 1965), 16, Les Presses de l’Université de Montréal, 1966 | Zbl

[51] Giorgio Talenti Best constant in Sobolev inequality, Ann. Mat. Pura Appl., Volume 110 (1976), pp. 353-372 | DOI | MR | Zbl

[52] Juan Luis Vázquez The porous medium equation. Mathematical theory, Oxford Mathematical Monographs, Clarendon Press, 2007, xxii+624 pages | MR

[53] Cédric Villani Optimal transport. Old and new., Springer, 2009, xxii+973 pages | DOI | Zbl

[54] Cédric Villani Inégalités isopérimétriques dans les espaces métriques mesurés [d’après F. Cavalletti & A. Mondino], Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120–1135 (Astérisque), Volume 407, Société Mathématique de France, 2019, pp. 213-265 | DOI | MR | Zbl

[55] Simon Zugmeyer Dynamical approaches to Sharp Sobolev inequalities, Ph. D. Thesis, Université de Lyon (France) (2019) (https://tel.archives-ouvertes.fr/tel-02611806)

[56] Simon Zugmeyer Entropy flows and functional inequalities in convex sets (2020) | arXiv

Cited by Sources: