June 2017, 37(6): 3059-3078. doi: 10.3934/dcds.2017131

Onofri inequalities and rigidity results

1. 

Ceremade, CNRS UMR n$^{o}$ 7534 and Université Paris-Dauphine PSL research university Place de Lattre de Tassigny, 75775 Paris Cédex 16, France

2. 

RICAM and Universität Wien Wolfgang Pauli Institute Oskar-Morgenstern-Platz 1,1090 Wien, Austria

* Corresponding author: Jean Dolbeault

Received  November 2015 Revised  January 2017 Published  February 2017

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

Citation: Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Onofri inequalities and rigidity results. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3059-3078. doi: 10.3934/dcds.2017131
References:
[1]

T. Aubin, Meilleures constantes dans le théoréme d'inclusion de Sobolev et un théoréme de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal., 32 (1979), 148-174. doi: 10.1016/0022-1236(79)90052-1.

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations vol. 252 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[3]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270. doi: 10.1215/S0012-7094-96-08511-7.

[4]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[5]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242. doi: 10.2307/2946638.

[6]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187-190.

[7]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin, 1971.

[8]

R. J. Berman and B. Berndtsson, Moser-Trudinger type inequalities for complex Monge-Ampére operators and Aubin's "hypothése fondamentale", ArXiv e-prints.

[9]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922.

[10]

P. BilerL. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, Journal of Mathematical Biology, 63 (2011), 1-32. doi: 10.1007/s00285-010-0357-5.

[11]

P. BilerG. KarchP. Laurençot and T. Nadzieja, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), 1563-1583. doi: 10.1002/mma.743.

[12]

J. Campos and J. Dolbeault, A functional framework for the Keller-Segel system: Logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities, C. R. Math. Acad. Sci. Paris, 350 (2012), 949-954. doi: 10.1016/j.crma.2012.10.023.

[13]

J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic keller-segel model in the plane, Comm. Partial Differential Equations, 39 (2014), 806-841. doi: 10.1080/03605302.2014.885046.

[14]

E. A. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on ${\mathbb{S}^n} $, Geom. Funct. Anal., 2 (1992), 90-104. doi: 10.1007/BF01895706.

[15]

S.-Y. A. Chang, Extremal functions in a sharp form of Sobolev inequality, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, (1987), 715-723.

[16]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on ${\mathbb{S}^2} $, J. Differential Geom., 27 (1988), 259-296.

[17]

S. -Y. A. Chang, Non-linear Elliptic Equations in Conformal Geometry Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/006.

[18]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on ${\mathbb{S}^2} $, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560.

[19]

S. -Y. A. Chang and P. C. Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math., 56 (2003), 1135-1150, Dedicated to the memory of Jürgen K. Moser. doi: 10.1002/cpa.3029.

[20]

W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[21]

P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math. (2), 103 (1979), 353-374.

[22]

J. Demange, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., 254 (2008), 593-611. doi: 10.1016/j.jfa.2007.01.017.

[23]

J. Dolbeault and M. J. Esteban, Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, 27 (2014), 435-465. doi: 10.1088/0951-7715/27/3/435.

[24]

J. DolbeaultM. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri inequality, Chinese Annals of Math. B, 36 (2015), 777-802. doi: 10.1007/s11401-015-0976-7.

[25]

J. DolbeaultM. J. EstebanM. Kowalczyk and M. Loss, Improved interpolation inequalities on the sphere, Discrete and Continuous Dynamical Systems Series S (DCDS-S), 7 (2014), 695-724. doi: 10.3934/dcdss.2014.7.695.

[26]

J. DolbeaultM. J. Esteban and M. Loss, Nonlinear flows and rigidity results on compact manifolds, Journal of Functional Analysis, 267 (2014), 1338-1363. doi: 10.1016/j.jfa.2014.05.021.

[27]

Z. Faget, Best constants in the exceptional case of Sobolev inequalities, Math. Z., 252 (2006), 133-146. doi: 10.1007/s00209-005-0850-5.

[28]

Z. Faget, Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 360 (2008), 2303-2325. doi: 10.1090/S0002-9947-07-04308-5.

[29]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphéres, Bull. Sci. Math., 121 (1997), 71-96.

[30]

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

[31]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications vol. 187 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.

[32]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^n} $, in Mathematical analysis and applications, Part A, vol. 7 of Adv. in Math. Suppl. Stud., Academic Press, New York-London, (1981), 369-402.

[33]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities vol. 5 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 1999.

[34]

C. W. Hong, A best constant and the Gaussian curvature, Proc. Amer. Math. Soc., 97 (1986), 737-747. doi: 10.1090/S0002-9939-1986-0845999-7.

[35]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536.

[36]

A. Lichnerowicz, Géométrie des Groupes de Transformations Travaux et Recherches Mathématiques, Ⅲ. Dunod, Paris, 1958.

[37]

J. R. Licois and L. Véron, Un théoréme d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337-1342.

[38]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[39]

Y. NaitoT. Suzuki and K. Yoshida, Self-similar solutions to a parabolic system modeling chemotaxis, J. Differential Equations, 184 (2002), 386-421. doi: 10.1006/jdeq.2001.4146.

[40]

M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal., 145 (1998), 161-195. doi: 10.1007/s002050050127.

[41]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. doi: 10.1007/BF01212171.

[42]

B. OsgoodR. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211. doi: 10.1016/0022-1236(88)90070-5.

[43]

D. H. PhongJ. SongJ. Sturm and B. Weinkove, The Moser-Trudinger inequality on Kähler-Einstein manifolds, American Journal of Mathematics, 130 (2008), 1067-1085. doi: 10.1353/ajm.0.0013.

[44]

Y. A. Rubinstein, On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood, J. Funct. Anal., 255 (2008), 2641-2660. doi: 10.1016/j.jfa.2007.10.009.

[45]

Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math., 218 (2008), 1526-1565. doi: 10.1016/j.aim.2008.03.017.

[46]

G. Tarantello, Selfdual Gauge Field Vortices: An Analytical Approach Progress in Nonlinear Differential Equations and their Applications, 72, Birkhäuser Boston, Inc. , Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0.

[47]

G. Tian and X. Zhu, A nonlinear inequality of Moser-Trudinger type, Calculus of Variations and Partial Differential Equations, 10 (2000), 349-354. doi: 10.1007/s005260010349.

[48]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

show all references

References:
[1]

T. Aubin, Meilleures constantes dans le théoréme d'inclusion de Sobolev et un théoréme de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal., 32 (1979), 148-174. doi: 10.1016/0022-1236(79)90052-1.

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations vol. 252 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[3]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270. doi: 10.1215/S0012-7094-96-08511-7.

[4]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[5]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242. doi: 10.2307/2946638.

[6]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187-190.

[7]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin, 1971.

[8]

R. J. Berman and B. Berndtsson, Moser-Trudinger type inequalities for complex Monge-Ampére operators and Aubin's "hypothése fondamentale", ArXiv e-prints.

[9]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922.

[10]

P. BilerL. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, Journal of Mathematical Biology, 63 (2011), 1-32. doi: 10.1007/s00285-010-0357-5.

[11]

P. BilerG. KarchP. Laurençot and T. Nadzieja, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), 1563-1583. doi: 10.1002/mma.743.

[12]

J. Campos and J. Dolbeault, A functional framework for the Keller-Segel system: Logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities, C. R. Math. Acad. Sci. Paris, 350 (2012), 949-954. doi: 10.1016/j.crma.2012.10.023.

[13]

J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic keller-segel model in the plane, Comm. Partial Differential Equations, 39 (2014), 806-841. doi: 10.1080/03605302.2014.885046.

[14]

E. A. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on ${\mathbb{S}^n} $, Geom. Funct. Anal., 2 (1992), 90-104. doi: 10.1007/BF01895706.

[15]

S.-Y. A. Chang, Extremal functions in a sharp form of Sobolev inequality, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, (1987), 715-723.

[16]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on ${\mathbb{S}^2} $, J. Differential Geom., 27 (1988), 259-296.

[17]

S. -Y. A. Chang, Non-linear Elliptic Equations in Conformal Geometry Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/006.

[18]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on ${\mathbb{S}^2} $, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560.

[19]

S. -Y. A. Chang and P. C. Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math., 56 (2003), 1135-1150, Dedicated to the memory of Jürgen K. Moser. doi: 10.1002/cpa.3029.

[20]

W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[21]

P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math. (2), 103 (1979), 353-374.

[22]

J. Demange, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., 254 (2008), 593-611. doi: 10.1016/j.jfa.2007.01.017.

[23]

J. Dolbeault and M. J. Esteban, Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, 27 (2014), 435-465. doi: 10.1088/0951-7715/27/3/435.

[24]

J. DolbeaultM. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri inequality, Chinese Annals of Math. B, 36 (2015), 777-802. doi: 10.1007/s11401-015-0976-7.

[25]

J. DolbeaultM. J. EstebanM. Kowalczyk and M. Loss, Improved interpolation inequalities on the sphere, Discrete and Continuous Dynamical Systems Series S (DCDS-S), 7 (2014), 695-724. doi: 10.3934/dcdss.2014.7.695.

[26]

J. DolbeaultM. J. Esteban and M. Loss, Nonlinear flows and rigidity results on compact manifolds, Journal of Functional Analysis, 267 (2014), 1338-1363. doi: 10.1016/j.jfa.2014.05.021.

[27]

Z. Faget, Best constants in the exceptional case of Sobolev inequalities, Math. Z., 252 (2006), 133-146. doi: 10.1007/s00209-005-0850-5.

[28]

Z. Faget, Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 360 (2008), 2303-2325. doi: 10.1090/S0002-9947-07-04308-5.

[29]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphéres, Bull. Sci. Math., 121 (1997), 71-96.

[30]

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

[31]

N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications vol. 187 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/187.

[32]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^n} $, in Mathematical analysis and applications, Part A, vol. 7 of Adv. in Math. Suppl. Stud., Academic Press, New York-London, (1981), 369-402.

[33]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities vol. 5 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 1999.

[34]

C. W. Hong, A best constant and the Gaussian curvature, Proc. Amer. Math. Soc., 97 (1986), 737-747. doi: 10.1090/S0002-9939-1986-0845999-7.

[35]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536.

[36]

A. Lichnerowicz, Géométrie des Groupes de Transformations Travaux et Recherches Mathématiques, Ⅲ. Dunod, Paris, 1958.

[37]

J. R. Licois and L. Véron, Un théoréme d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337-1342.

[38]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[39]

Y. NaitoT. Suzuki and K. Yoshida, Self-similar solutions to a parabolic system modeling chemotaxis, J. Differential Equations, 184 (2002), 386-421. doi: 10.1006/jdeq.2001.4146.

[40]

M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal., 145 (1998), 161-195. doi: 10.1007/s002050050127.

[41]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. doi: 10.1007/BF01212171.

[42]

B. OsgoodR. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211. doi: 10.1016/0022-1236(88)90070-5.

[43]

D. H. PhongJ. SongJ. Sturm and B. Weinkove, The Moser-Trudinger inequality on Kähler-Einstein manifolds, American Journal of Mathematics, 130 (2008), 1067-1085. doi: 10.1353/ajm.0.0013.

[44]

Y. A. Rubinstein, On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood, J. Funct. Anal., 255 (2008), 2641-2660. doi: 10.1016/j.jfa.2007.10.009.

[45]

Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math., 218 (2008), 1526-1565. doi: 10.1016/j.aim.2008.03.017.

[46]

G. Tarantello, Selfdual Gauge Field Vortices: An Analytical Approach Progress in Nonlinear Differential Equations and their Applications, 72, Birkhäuser Boston, Inc. , Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0.

[47]

G. Tian and X. Zhu, A nonlinear inequality of Moser-Trudinger type, Calculus of Variations and Partial Differential Equations, 10 (2000), 349-354. doi: 10.1007/s005260010349.

[48]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

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