doi: 10.3934/dcds.2022080
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Parabolic methods for ultraspherical interpolation inequalities

1. 

Ceremade, UMR CNRS n° 7534, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

2. 

School of Mathematical Sciences, Beihang University, No. 37 Xueyuan Road, Haidian, Beijing 100191, China

* Corresponding author: An Zhang

It is with great pleasure that we dedicate this paper to Juan Luis Vázquez on the occasion of his 75th birthday.

Received  February 2022 Early access June 2022

The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.

Citation: Jean Dolbeault, An Zhang. Parabolic methods for ultraspherical interpolation inequalities. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022080
References:
[1]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778. 

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, 1983/84, vol. 1123 of Lecture Notes in Math., Springer, Berlin, 1985,177–206, URL http://www.numdam.org/item/?id=SPS_1985__19__177_0. doi: 10.1007/BFb0075847.

[3]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413. 

[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.-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.

[8]

M. Bonforte, J. Dolbeault, B. Nazaret and N. Simonov, Stability in Gagliardo-Nirenberg-Sobolev inequalities, Preprint hal-02887010 and arXiv: 2007.03674, to appear in Memoirs AMS.

[9]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82.  doi: 10.1007/s006050170032.

[10]

J. A. Carrillo and G. Toscani, Asymptotic $\mathrm L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana University Mathematics Journal, 49 (2000), 113-142.  doi: 10.1512/iumj.2000.49.1756.

[11]

J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations, Communications in Partial Differential Equations, 28 (2003), 1023-1056.  doi: 10.1081/PDE-120021185.

[12]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Communications on Pure and Applied Mathematics, 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[13]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332.  doi: 10.1016/S0001-8708(03)00080-X.

[14]

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.

[15]

J. Dolbeault, Functional inequalities: Nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results, Milan Journal of Mathematics, 89 (2021), 355-386.  doi: 10.1007/s00032-021-00341-y.

[16]

J. Dolbeault and M. J. Esteban, Improved interpolation inequalities and stability, Advanced Nonlinear Studies, 20 (2020), 277-291.  doi: 10.1515/ans-2020-2080.

[17]

J. DolbeaultM. J. EstebanM. Kowalczyk and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences, Chinese Annals of Mathematics, Series B, 34 (2013), 99-112.  doi: 10.1007/s11401-012-0756-6.

[18]

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.

[19]

J. DolbeaultM. J. EstebanA. Laptev and M. Loss, One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: Remarks on duality and flows, Journal of the London Mathematical Society, 90 (2014), 525-550.  doi: 10.1112/jlms/jdu040.

[20]

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.

[21]

J. DolbeaultM. J. Esteban and M. Loss, Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization, Journal of Elliptic and Parabolic Equations, 2 (2016), 267-295.  doi: 10.1007/BF03377405.

[22]

J. DolbeaultM. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., 206 (2016), 397-440.  doi: 10.1007/s00222-016-0656-6.

[23]

J. Dolbeault, M. J. Esteban and M. Loss, Interpolation inequalities on the sphere: Linear vs. nonlinear flows (Inégalités d'interpolation sur la sphère : Flots non-linéaires vs. flots linéaires), Annales de la Faculté des Sciences de Toulouse Sér. 6, 26 (2017), 351–379. doi: 10.5802/afst.1536.

[24]

J. DolbeaultM. J. Esteban and M. Loss, Symmetry and symmetry breaking: Rigidity and flows in elliptic PDEs, Proc. Int. Cong. of Math. 2018, Rio de Janeiro, 3 (2018), 2261-2285.  doi: 10.1142/9789813272880_0138.

[25]

J. DolbeaultM. J. EstebanM. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities, Comptes Rendus Mathématique, 355 (2017), 133-154.  doi: 10.1016/j.crma.2017.01.004.

[26]

L. Dupaigne, I. Gentil and S. Zugmeyer, A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality, Preprint arXiv: 2111.15383, 2021.

[27]

V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Journal of Differential Equations, 191 (2003), 121-142.  doi: 10.1016/S0022-0396(02)00085-2.

[28]

R. L. Frank, Degenerate stability of some Sobolev inequalities, Annales de l'Institut Henri Poincaré C Analyse Non Linéaire, (2022). Preprint arXiv: 2107.11608, 2021. doi: 10.4171/aihpc/35.

[29]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[30]

I. Gentil and S. Zugmeyer, A family of Beckner inequalities under various curvature-dimension conditions, Bernoulli, 27 (2021), 751-771.  doi: 10.3150/20-bej1228.

[31]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[32]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349–374, URL http://doi.org/10.1007/978-3-642-55925-9_43. doi: 10.2307/2007032.

[33]

C. E. Mueller and F. B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere, J. Funct. Anal., 48 (1982), 252-283.  doi: 10.1016/0022-1236(82)90069-6.

[34]

J. M. Pearson, Best constants in Sobolev inequalities for ultraspherical polynomials, Arch. Rational Mech. Anal., 116 (1992), 361-374.  doi: 10.1007/BF00375673.

[35]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006, Equations of porous medium type. doi: 10.1093/acprof:oso/9780199202973.001.0001.

[36]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. doi: 10.1093/acprof:oso/9780198569039.001.0001.

show all references

References:
[1]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778. 

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, 1983/84, vol. 1123 of Lecture Notes in Math., Springer, Berlin, 1985,177–206, URL http://www.numdam.org/item/?id=SPS_1985__19__177_0. doi: 10.1007/BFb0075847.

[3]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413. 

[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.-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.

[8]

M. Bonforte, J. Dolbeault, B. Nazaret and N. Simonov, Stability in Gagliardo-Nirenberg-Sobolev inequalities, Preprint hal-02887010 and arXiv: 2007.03674, to appear in Memoirs AMS.

[9]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82.  doi: 10.1007/s006050170032.

[10]

J. A. Carrillo and G. Toscani, Asymptotic $\mathrm L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana University Mathematics Journal, 49 (2000), 113-142.  doi: 10.1512/iumj.2000.49.1756.

[11]

J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations, Communications in Partial Differential Equations, 28 (2003), 1023-1056.  doi: 10.1081/PDE-120021185.

[12]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Communications on Pure and Applied Mathematics, 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[13]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332.  doi: 10.1016/S0001-8708(03)00080-X.

[14]

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.

[15]

J. Dolbeault, Functional inequalities: Nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results, Milan Journal of Mathematics, 89 (2021), 355-386.  doi: 10.1007/s00032-021-00341-y.

[16]

J. Dolbeault and M. J. Esteban, Improved interpolation inequalities and stability, Advanced Nonlinear Studies, 20 (2020), 277-291.  doi: 10.1515/ans-2020-2080.

[17]

J. DolbeaultM. J. EstebanM. Kowalczyk and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences, Chinese Annals of Mathematics, Series B, 34 (2013), 99-112.  doi: 10.1007/s11401-012-0756-6.

[18]

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.

[19]

J. DolbeaultM. J. EstebanA. Laptev and M. Loss, One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: Remarks on duality and flows, Journal of the London Mathematical Society, 90 (2014), 525-550.  doi: 10.1112/jlms/jdu040.

[20]

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.

[21]

J. DolbeaultM. J. Esteban and M. Loss, Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization, Journal of Elliptic and Parabolic Equations, 2 (2016), 267-295.  doi: 10.1007/BF03377405.

[22]

J. DolbeaultM. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., 206 (2016), 397-440.  doi: 10.1007/s00222-016-0656-6.

[23]

J. Dolbeault, M. J. Esteban and M. Loss, Interpolation inequalities on the sphere: Linear vs. nonlinear flows (Inégalités d'interpolation sur la sphère : Flots non-linéaires vs. flots linéaires), Annales de la Faculté des Sciences de Toulouse Sér. 6, 26 (2017), 351–379. doi: 10.5802/afst.1536.

[24]

J. DolbeaultM. J. Esteban and M. Loss, Symmetry and symmetry breaking: Rigidity and flows in elliptic PDEs, Proc. Int. Cong. of Math. 2018, Rio de Janeiro, 3 (2018), 2261-2285.  doi: 10.1142/9789813272880_0138.

[25]

J. DolbeaultM. J. EstebanM. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities, Comptes Rendus Mathématique, 355 (2017), 133-154.  doi: 10.1016/j.crma.2017.01.004.

[26]

L. Dupaigne, I. Gentil and S. Zugmeyer, A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality, Preprint arXiv: 2111.15383, 2021.

[27]

V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Journal of Differential Equations, 191 (2003), 121-142.  doi: 10.1016/S0022-0396(02)00085-2.

[28]

R. L. Frank, Degenerate stability of some Sobolev inequalities, Annales de l'Institut Henri Poincaré C Analyse Non Linéaire, (2022). Preprint arXiv: 2107.11608, 2021. doi: 10.4171/aihpc/35.

[29]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[30]

I. Gentil and S. Zugmeyer, A family of Beckner inequalities under various curvature-dimension conditions, Bernoulli, 27 (2021), 751-771.  doi: 10.3150/20-bej1228.

[31]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[32]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349–374, URL http://doi.org/10.1007/978-3-642-55925-9_43. doi: 10.2307/2007032.

[33]

C. E. Mueller and F. B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere, J. Funct. Anal., 48 (1982), 252-283.  doi: 10.1016/0022-1236(82)90069-6.

[34]

J. M. Pearson, Best constants in Sobolev inequalities for ultraspherical polynomials, Arch. Rational Mech. Anal., 116 (1992), 361-374.  doi: 10.1007/BF00375673.

[35]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006, Equations of porous medium type. doi: 10.1093/acprof:oso/9780199202973.001.0001.

[36]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. doi: 10.1093/acprof:oso/9780198569039.001.0001.

Figure 1.  Admissible ranges of $ m $ for $ n = 0.25 $, $ 1 $, $ 1.8 $ (first line, from the left to the right) and $ n = 2 $, $ 3 $, $ 4 $ (second line). The dotted and dashed curves correspond respectively to $ m = n/(n+2) $ and $ m = (n-2)/n $. In Lemma 4, Condition (21) amounts to $ (n-2)/n<m<1 $ and $ m\neq n/(n+2) $
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