Interpolation inequalities in <inline-formula><tex-math id="M1">\begin{document}$ \mathrm W^{1,p}( {\mathbb S}^1) $\end{document}</tex-math></inline-formula> and <i>carré du champ</i> methods

Jean Dolbeault; Marta García-Huidobro; Rául Manásevich

doi:10.3934/dcds.2020014

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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

Jean Dolbeault ^1,, , Marta García-Huidobro ^2, and Rául Manásevich ^3,

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 $.

Keywords: Interpolation, Gagliardo-Nirenberg inequalities, bifurcation, branches of solutions, elliptic equations, p-Laplacian, entropy, Fisher information, carré du champ method, Poincaré inequality, rigidity, uniqueness, rescaling, period, nonlinear Keller-Lieb-Thirring energy estimates.

Mathematics Subject Classification: Primary: 35J92, 35K92; Secondary: 49K15, 58J35.

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

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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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.

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

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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 $

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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

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Figure 4. The curves $ p\mapsto\lambda_1 $ (dotted) and $ p\mapsto\lambda_1^\star $ (plain) differ

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American Institute of Mathematical Sciences

Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods

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Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods

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