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January  2020, 40(1): 375-394. doi: 10.3934/dcds.2020014

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

* Corresponding author: Jean Dolbeault

Received  February 2019 Revised  June 2019 Published  October 2019

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 $.

Citation: Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014
References:
[1]

A. ArnoldJ. A. CarrilloL. DesvillettesJ. DolbeaultA. JüngelC. LedermanP. A. MarkowichG. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

J. DolbeaultM. J. Esteban and A. Laptev, Spectral estimates on the sphere, Analysis & PDE, 7 (2014), 435-460.  doi: 10.2140/apde.2014.7.435.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. ArnoldJ. A. CarrilloL. DesvillettesJ. DolbeaultA. JüngelC. LedermanP. A. MarkowichG. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

J. DolbeaultM. J. Esteban and A. Laptev, Spectral estimates on the sphere, Analysis & PDE, 7 (2014), 435-460.  doi: 10.2140/apde.2014.7.435.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

Figure 1.  The vector field $ (X,Y)\mapsto(|Y|^{\frac p{p-1}-2}\,Y,|X|^{p-2}\,X-|X|^{q-2}\,X) $ and periodic trajectories corresponding to $ a = 1.35 $ (with positive $ X $) and $ a = 1.8 $ (with sign-changing $ X $) are shown for $ p = 2.5 $ and $ q = 3 $. The zero-energy level is also shown
Figure 2.  The period $ T_a $ of the solution of (12) with initial datum $ X(0) = a\in(0,1) $ and $ Y(0) = 0 $ as a function of $ a $ for $ p = 3 $ and $ q = 5 $. We observe that $ \lim_{a\to0}T_a = +\infty $ and $ \lim_{a\to1}T_a = 0 $
Figure 3.  Left: the branch $ \lambda\mapsto\mu(\lambda) $ for $ p = 3 $ and $ q = 5 $. Right: the curve $ \lambda\mapsto\lambda-\mu(\lambda) $. In both cases, the bifurcation point $ \lambda = \lambda_1^\star $ is shown by a vertical line
Figure 4.  The curves $ p\mapsto\lambda_1 $ (dotted) and $ p\mapsto\lambda_1^\star $ (plain) differ
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