The Moser-Trudinger-Onofri inequality

Abstract

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.

Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

Keywords

Moser-Trudinger-Onofri inequality / Duality / Mass transportation / Fast diffusion equation / Rigidity

References

[1] Adachi S., Tanaka K.. Trudinger type inequalities in RNand their best exponents. Proc. Amer. Math. Soc., 2000, 128(7): 2051-2057

[2] Arnold A., Markowich P., Toscani G., Unterreiter A.. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Differential Equations, 2001, 26(1–2): 43-100

[3] Aubin T.. Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geometry, 1976, 11(4): 573-598

[4] Aubin T.. Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal., 1979, 32(2): 148-174

[5] Baernstein A. I.. A unified approach to symmetrization. Partial Differential Equations of Elliptic Type, 1994, Cambridge: Cambridge Univ. Press 47-91

[6] Baernstein A. I., Taylor B. A.. Spherical rearrangements, subharmonic functions, and *-functions in n-space. Duke Math. J., 1976, 43(2): 245-268

[7] Bakry D., émery M.. Hypercontractivité de semi-groupes de diffusion. C. R. Acad. Sci. Paris Sér. I Math., 1984, 299(15): 775-778

[8] Beckner W.. Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2), 1993, 138(1): 213-242

[9] Bentaleb A.. Inégalité de Sobolev pour l’opérateur ultrasphérique. C. R. Acad. Sci. Paris Sér. I Math., 1993, 317(2): 187-190

[10] Bidaut-Véron M.-F., Véron L.. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math., 1991, 106(3): 489-539

[11] Blanchet A., Carlen E. A., Carrillo J. A.. Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model. J. Funct. Anal., 2012, 262(5): 2142-2230

[12] Bliss G.. An integral inequality. Journal of the London Mathematical Society, 1930, 1(1): 40

[13] Branson T., Fontana L., Morpurgo C.. Moser-Trudinger and Beckner-Onofris inequalities on the CRsphere. Annals of Mathematics, 2013, 177: 1-52

[14] Brothers J. E., Ziemer W. P.. Minimal rearrangements of Sobolev functions. J. Reine Angew. Math., 1988, 384: 153-179

[15] Caffarelli L., Kohn R., Nirenberg L.. First order interpolation inequalities with weights. Compositio Math., 1984, 53(3): 259-275

[16] Calvez V., Corrias L.. The parabolic-parabolic Keller-Segel model in ℝ². Commun. Math. Sci., 2008, 6(2): 417-447

[17] Carlen E. A., Carrillo J. A., Loss M.. Hardy-Littlewood-Sobolev inequalities via fast diffusion flows. Proc. Natl. Acad. Sci. USA, 2010, 107(46): 19696-19701

[18] Carlen E. A., Figalli A.. Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation. Duke Math. J., 2013, 162(3): 579-625

[19] Carlen E. A., Loss M.. Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on S ⁿ. Geom. Funct. Anal., 1992, 2(1): 90-104

[20] Carleson L., Chang S. Y. A.. On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math, 1986, 110(2): 113-127

[21] Carrillo J. A., Jüngel A., Markowich P. A. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math., 2001, 133(1): 1-82

[22] Carrillo J. A., Toscani G.. Asymptotic L¹-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J., 2000, 49(1): 113-142

[23] Chang S. Y. A.. Extremal functions in a sharp form of Sobolev inequality. Proceedings of the International Congress of Mathematicians, 1987 715-723

[24] Chang S. Y. A., Yang P. C.. Prescribing Gaussian curvature on S ². Acta Math., 1987, 159(3–4): 215-259

[25] Chang S. Y. A., Yang P. C.. Conformal deformation of metrics on S ². J. Differential Geom., 1988, 27(2): 259-296

[26] Chang S. Y. A., Yang P. C.. The inequality of Moser and Trudinger and applications to conformal geometry, dedicated to the memory of Jürgen K. Moser). Comm. Pure Appl. Math., 2003, 56(8): 1135-1150

[27] Cordero-Erausquin D., Nazaret B., Villani C.. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math., 2004, 182(2): 307-332

[28] Del Pino M., Dolbeault J.. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl, 2002, 81(9): 847-875

[29] Del Pino M., Dolbeault J.. The Euclidean Onofri inequality in higher dimensions. Int. Math. Res. Not. IMRN, 2013, 15: 3600-3611

[30] Dolbeault J.. Sobolev and Hardy-Littlewood-Sobolev inequalities: Duality and fast diffusion. Math. Res. Lett., 2011, 18(6): 1037-1050

[31] Dolbeault J., Esteban M. J., Kowalczyk M., Loss M.. Sharp interpolation inequalities on the sphere: New methods and consequences. Chin. Ann. Math., 2013, 34(1): 99-112

[32] Dolbeault J., Esteban M. J., Kowalczyk M., Loss M.. Improved interpolation inequalities on the sphere. Discrete and Continuous Dynamical Systems Series S (DCDS-S), 2014, 7(4): 695-724

[33] Dolbeault J., Esteban M. J., Laptev A.. Spectral estimates on the sphere. Analysis PDE, 2014, 7(2): 435-460

[34] Dolbeault J., Esteban M. J., Laptev A., Loss M.. One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows. Journal of the London Mathematical Society, 2014, 90(2): 525-550

[35] Dolbeault J., Esteban M. J., Laptev A., Loss M.. Spectral properties of Schrödinger operators on compact manifolds: Rigidity, flows, interpolation and spectral estimates. Comptes Rendus Mathématique, 2013, 351(11): 437-440

[36] Dolbeault J., Esteban M. J., Loss M.. Nonlinear flows and rigidity results on compact manifolds. J. Funct. Anal., 2014, 267(5): 1338-1363

[37] Dolbeault J., Esteban M. J., Tarantello G.. The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2008, 7(2): 313-341

[38] Dolbeault J., Esteban M. J., Tarantello G.. Multiplicity results for the assigned Gauss curvature problem in ℝ². Nonlinear Anal., 2009, 70(8): 2870-2881

[39] Dolbeault J., Jankowiak G.. Sobolev and Hardy-Littlewood-Sobolev inequalities. J. Differential Equations, 2014, 257(6): 1689-1720

[40] Dolbeault J., Toscani G.. Improved interpolation inequalities, relative entropy and fast diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2013, 30(5): 917-934

[41] Flucher M.. Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv., 1992, 67(3): 471-497

[42] Fontenas É.. Sur les constantes de Sobolev des variétés Riemanniennes compactes et les fonctions extrémales des sphères. Bull. Sci. Math., 1997, 121(2): 71-96

[43] Fontenas É.. Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre. Séminaire de Probabilités, 1998, 1686: 14-29

[44] Gajewski H., Zacharias K.. Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr., 1998, 195: 77-114

[45] Ghigi A.. On the Moser-Onofri and Prékopa-Leindler inequalities. Collect. Math., 2005, 56(2): 143-156

[46] Ghoussoub N., Lin C.-S.. On the best constant in the Moser-Onofri-Aubin inequality. Comm. Math. Phys., 2010, 298(3): 869-878

[47] Ghoussoub, N. and Moradifam, A., Functional inequalities: New perspectives and new applications, Mathematical Surveys and Monographs, Vol. 187, A. M. S., Providence, RI, 2013. ISBN 978-0-8218-9152-0

[48] Gidas B., Spruck J.. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 1981, 34(4): 525-598

[49] Hersch J.. Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A–B, 1970, 270: A1645-A1648

[50] Hong C. W.. A best constant and the Gaussian curvature. Proc. Amer. Math. Soc., 1986, 97(4): 737-747

[51] Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., Vol. 1150, Springer-Verlag, Berlin, 1985. ISBN 3-540-15693-3

[52] Lam N., Lu G.. A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument. J. Differential Equations, 2013, 255(3): 298-325

[53] DIFaddLieb E. H.. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math, 1983, 118(2): 349-374

[54] Lieb E. H., Loss M.. Analysis. Graduate Studies in Mathematics, 2001

[55] McCann R. J.. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J., 1995, 80(2): 309-323

[56] McCann R. J.. A convexity principle for interacting gases. Adv. Math., 1997, 128(1): 153-179

[57] DIFadd McLeod J. B., Peletier L. A.. Observations on Moser’s inequality. Arch. Rational Mech. Anal., 1989, 106(3): 261-285

[58] Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 107, 7–1092, 1970–1971.

[59] Newman W. I.. A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity I. J.Math. Phys., 1984, 25(10): 3120-3123

[60] Okikiolu K.. Extremals for logarithmic Hardy-Littlewood-Sobolev inequalities on compact manifolds. Geom. Funct. Anal., 2008, 17(5): 1655-1684

[61] Onofri E.. On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys., 1982, 86(3): 321-326

[62] Osgood B., Phillips R., Sarnak P.. Extremals of determinants of Laplacians. J. Funct. Anal., 1988, 80(1): 148-211

[63] Ralston J.. A Lyapunov functional for the evolution of solutions to the porous medium equation to selfsimilarity II. J. Math. Phys., 1984, 25(10): 3124-3127

[64] Rosen, G., Minimum value for c in the Sobolev inequality ∥ϕ ³∥ ≤ c ∥▽ϕ∥³, SIAM J. Appl. Math., 21, 1971, 30–32.

[65] Rubinstein Y. A.. On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood. J. Funct. Anal., 2008, 255(9): 2641-2660

[66] Rubinstein Y. A.. Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics. Adv. Math., 2008, 218(5): 1526-1565

[67] Talenti G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl, 1976, 110: 353-372

[68] Trudinger N.. On imbeddings into orlicz spaces and some applications. Indiana Univ. Math. J., 1968, 17: 473-483

[69] Villani, C., Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin, 2009. ISBN 978-3-540-71049-3. DOI: 10.1007/978-3-540-71050-9