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Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods
1. | CEREMADE (CNRS UMR n° 7534), PSL university, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France |
2. | Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago de Chile, Chile |
3. | DIM & CMM (UMI CNRS n° 2071), FCFM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile |
This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in $ \mathrm W^{1,p}( {\mathbb S}^1) $ with $ p\ge2 $. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a $ p $-Laplacian type operator. It is remarkable that the carré du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever $ p\neq2 $.
References:
[1] |
A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani,
Entropies and equilibria of many-particle systems: An essay on recent research, Monatsh. Math., 142 (2004), 35-43.
doi: 10.1007/s00605-004-0239-2. |
[2] |
A. Arnold and J. Dolbeault,
Refined convex Sobolev inequalities, J. Funct. Anal., 225 (2005), 337-351.
doi: 10.1016/j.jfa.2005.05.003. |
[3] |
D. Bakry and M. Émery,
Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778.
|
[4] |
D. Bakry and M. Émery, Diffusions hypercontractives, In: Séminaire de Probabilités, XIX, 1983/84, Lecture Notes in Math., 1123 (1985), 177–206. Springer, Berlin.
doi: 10.1007/BFb0075847. |
[5] |
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. |
[6] |
M. Del Pino and J. Dolbeault,
The optimal Euclidean $\mathrm L^p$-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151-161.
doi: 10.1016/S0022-1236(02)00070-8. |
[7] |
J. Dolbeault, M. J. Esteban, M. 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. |
[8] |
J. Dolbeault, M. J. Esteban and A. Laptev,
Spectral estimates on the sphere, Analysis & PDE, 7 (2014), 435-460.
doi: 10.2140/apde.2014.7.435. |
[9] |
J. Dolbeault and M. Kowalczyk, Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates, Annales de la Faculté des sciences de Toulouse Mathématiques, 26 (2017), 949–977.
doi: 10.5802/afst.1557. |
[10] |
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. |
[11] |
R. Manásevich and J. Mawhin,
The spectrum of $p$-Laplacian systems with various boundary conditions and applications, Adv. Differential Equations, 5 (2000), 1289-1318.
|
[12] |
A. M. Matei,
First eigenvalue for the $p$-Laplace operator, Nonlinear Anal., 39 (2000), 1051-1068.
doi: 10.1016/S0362-546X(98)00266-1. |
[13] |
L. Véron,
Première valeur propre non nulle du $p$-laplacien et équations quasi linéaires elliptiques sur une variété riemannienne compacte, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 271-276.
|
[14] |
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
show all references
References:
[1] |
A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani,
Entropies and equilibria of many-particle systems: An essay on recent research, Monatsh. Math., 142 (2004), 35-43.
doi: 10.1007/s00605-004-0239-2. |
[2] |
A. Arnold and J. Dolbeault,
Refined convex Sobolev inequalities, J. Funct. Anal., 225 (2005), 337-351.
doi: 10.1016/j.jfa.2005.05.003. |
[3] |
D. Bakry and M. Émery,
Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778.
|
[4] |
D. Bakry and M. Émery, Diffusions hypercontractives, In: Séminaire de Probabilités, XIX, 1983/84, Lecture Notes in Math., 1123 (1985), 177–206. Springer, Berlin.
doi: 10.1007/BFb0075847. |
[5] |
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. |
[6] |
M. Del Pino and J. Dolbeault,
The optimal Euclidean $\mathrm L^p$-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151-161.
doi: 10.1016/S0022-1236(02)00070-8. |
[7] |
J. Dolbeault, M. J. Esteban, M. 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. |
[8] |
J. Dolbeault, M. J. Esteban and A. Laptev,
Spectral estimates on the sphere, Analysis & PDE, 7 (2014), 435-460.
doi: 10.2140/apde.2014.7.435. |
[9] |
J. Dolbeault and M. Kowalczyk, Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates, Annales de la Faculté des sciences de Toulouse Mathématiques, 26 (2017), 949–977.
doi: 10.5802/afst.1557. |
[10] |
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. |
[11] |
R. Manásevich and J. Mawhin,
The spectrum of $p$-Laplacian systems with various boundary conditions and applications, Adv. Differential Equations, 5 (2000), 1289-1318.
|
[12] |
A. M. Matei,
First eigenvalue for the $p$-Laplace operator, Nonlinear Anal., 39 (2000), 1051-1068.
doi: 10.1016/S0362-546X(98)00266-1. |
[13] |
L. Véron,
Première valeur propre non nulle du $p$-laplacien et équations quasi linéaires elliptiques sur une variété riemannienne compacte, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 271-276.
|
[14] |
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |



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