Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

Jean Dolbeault; Michał Kowalczyk

doi:10.5802/afst.1557

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates
[Unicité et rigidité pour des équations elliptiques non linéaires, inégalités d’interpolation et estimations spectrales]

Jean Dolbeault ¹ ; Michał Kowalczyk ²

¹ Ceremade, UMR CNRS n ∘ 7534, Université Paris-Dauphine, PSL research university, Pl. de Lattre de Tassigny, 75775 Paris Cédex 16, France
² Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI n ∘ 2807), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 949-977.

Résumé
Abstract

Cet article est consacré à la conjecture de Lin–Ni pour une équation semi-linéaire elliptique avec non-linéarité super-linéaire, sous-critique et des conditions de Neumann homogènes. Nous établissons un résultat de rigidité, c’est-à-dire nous prouvons que la seule solution positive est constante si le paramètre du problème est en dessous d’une borne explicite, reliée à la constante optimale d’une inégalité d’interpolation de Gagliardo–Nirenberg–Sobolev et aussi à une inégalité de Keller–Lieb–Thirring optimale. Nos résultats sont également valides dans un régime sous-linéaire. La borne de rigidité est obtenue par des méthodes de flots non-linéaires inspirées de résultats récents sur les variétés compactes, qui unifient des techniques d’équations elliptiques non-linéaires et la méthode du carré du champ en théorie des semi-groupes. Notre méthode requiert la convexité du domaine. Elle repose sur des quantités intégrales, prend en compte des estimations spectrales et fournit des inégalités améliorées.

This paper is devoted to the Lin–Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo–Nirenberg–Sobolev interpolation inequality and also with an optimal Keller–Lieb–Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carré du champ method in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.

Publié le : 2017-12-12

DOI : 10.5802/afst.1557

Classification : 35J60, 26D10, 46E35
Keywords: semilinear elliptic equations, Lin–Ni conjecture, Sobolev inequality, interpolation, Gagliardo–Nirenberg inequalities, Keller–Lieb–Thirring inequality, optimal constants, rigidity results, uniqueness, carré du champ method, CD($\rho $, $N$) condition, bifurcation, multiplicity, generalized entropy methods, heat flow, nonlinear diffusion, spectral gap inequality, Poincaré inequality, improved inequalities, non-Lipschitz nonlinearity, compact support principle
Mot clés : Inégalités fonctionnelles

Affiliations des auteurs :

Jean Dolbeault ¹ ; Michał Kowalczyk ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2017_6_26_4_949_0,
     author = {Jean Dolbeault and Micha{\l} Kowalczyk},
     title = {Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {949--977},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {4},
     year = {2017},
     doi = {10.5802/afst.1557},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1557/}
}

TY  - JOUR
AU  - Jean Dolbeault
AU  - Michał Kowalczyk
TI  - Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2017
SP  - 949
EP  - 977
VL  - 26
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1557/
DO  - 10.5802/afst.1557
LA  - en
ID  - AFST_2017_6_26_4_949_0
ER  -

%0 Journal Article
%A Jean Dolbeault
%A Michał Kowalczyk
%T Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2017
%P 949-977
%V 26
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1557/
%R 10.5802/afst.1557
%G en
%F AFST_2017_6_26_4_949_0

Jean Dolbeault; Michał Kowalczyk. Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 949-977. doi : 10.5802/afst.1557. https://afst.centre-mersenne.org/articles/10.5802/afst.1557/

Bibliographie
Cité par

[1] Anton Arnold; Jean-Philippe Bartier; Jean Dolbeault Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., Volume 5 (2007) no. 4, pp. 971-979 | DOI | Zbl

[2] Dominique Bakry; Michel Émery Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci., Paris, Sér. I, Volume 299 (1984) no. 15, pp. 775-778 | MR | Zbl

[3] Dominique Bakry; Michel Émery Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci., Paris, Sér. I, Volume 301 (1985) no. 8, pp. 411-413 | MR | Zbl

[4] Dominique Bakry; Michel Ledoux Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J., Volume 85 (1996) no. 1, pp. 253-270 | DOI | MR | Zbl

[5] Mikhaël Balabane; Jean Dolbeault; Hichem Ounaies Nodal solutions for a sublinear elliptic equation, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 52 (2003) no. 1, pp. 219-237 | DOI | MR | Zbl

[6] William Beckner A generalized Poincaré inequality for Gaussian measures, Proc. Am. Math. Soc., Volume 105 (1989) no. 2, pp. 397-400 | MR | Zbl

[7] Marie-Françoise Bidaut-Véron; Laurent Véron Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., Volume 106 (1991) no. 3, pp. 489-539 erratum in ibid. 112 (1993), no. 2, p. 445 | DOI | MR | Zbl

[8] Carmen Cortázar; Manuel Elgueta; Patricio Felmer Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Commun. Partial Differ. Equations, Volume 21 (1996) no. 3-4, pp. 507-520 | DOI | MR | Zbl

[9] Carmen Cortázar; Manuel Elgueta; Patricio Felmer Uniqueness of positive solutions of $Δ u + f (u) = 0$ in $ℝ^{N}, N \geq 3$ , Arch. Ration. Mech. Anal., Volume 142 (1998) no. 2, pp. 127-141 | DOI | MR | Zbl

[10] Jérôme Demange Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie, Université Paul Sabatier Toulouse 3 (France) (2005) (Ph. D. Thesis)

[11] Jérôme Demange Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., Volume 254 (2008) no. 3, pp. 593-611 | DOI | MR | Zbl

[12] Jean Dolbeault; Maria J. Esteban A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities, J. Numer. Math., Volume 20 (2013) no. 3-4, p. 233--249 | Zbl

[13] Jean Dolbeault; Maria J. Esteban Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, Volume 27 (2014) no. 3, pp. 435-465 http://stacks.iop.org/0951-7715/27/i=3/a=435 | DOI | Zbl

[14] Jean Dolbeault; Maria J. Esteban; Michal Kowalczyk; Michael Loss Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences, Chin. Ann. Math., Ser. B, Volume 34 (2013) no. 1, pp. 99-112 | DOI | Zbl

[15] Jean Dolbeault; Maria J. Esteban; Michal Kowalczyk; Michael Loss Improved interpolation inequalities on the sphere, Discrete Contin. Dyn. Syst., Ser. S, Volume 7 (2014) no. 4, pp. 695-724 | DOI | Zbl

[16] Jean Dolbeault; Maria J. Esteban; Ari Laptev Spectral estimates on the sphere, Anal. PDE, Volume 7 (2014) no. 2, pp. 435-460 | DOI | Zbl

[17] Jean Dolbeault; Maria J. Esteban; Ari Laptev; Michael Loss Spectral properties of Schrödinger operators on compact manifolds: Rigidity, flows, interpolation and spectral estimates, C. R., Math., Acad. Sci. Paris, Volume 351 (2013) no. 11–12, pp. 437 -440 | DOI | Zbl

[18] Jean Dolbeault; Maria J. Esteban; Ari Laptev; Michael Loss One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows, J. Lond. Math. Soc., Volume 90 (2014) no. 2, pp. 525-550 | DOI | Zbl

[19] Jean Dolbeault; Maria J. Esteban; Michael Loss Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., Volume 267 (2014) no. 5, pp. 1338 -1363 | DOI | Zbl

[20] Jean Dolbeault; Maria J. Esteban; Michael Loss Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders, C. R., Math., Acad. Sci. Paris, Volume 353 (2015) no. 9, pp. 813-818 | DOI | MR | Zbl

[21] Jean Dolbeault; Maria J. Esteban; Michael Loss Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., Volume 206 (2016) no. 2, pp. 397-440 | DOI | MR | Zbl

[22] Jean Dolbeault; Maria J. Esteban; Michael Loss Symmetry of optimizers of the Caffarelli-Kohn-Nirenberg inequalities, 2016 (https://hal.archives-ouvertes.fr/hal-01286546 and http://arxiv.org/abs/1603.03574)

[23] Jean Dolbeault; Maria J. Esteban; Michael Loss Interpolation inequalities on the sphere: linear vs. nonlinear flows, Ann. Fac. Sci. Toulouse, Volume 26 (2017) no. 2, pp. 351-379 | DOI | Zbl

[24] Jean Dolbeault; Maria J. Esteban; Gabriella Tarantello; Achilles Tertikas Radial symmetry and symmetry breaking for some interpolation inequalities, Calc. Var. Partial Differ. Equ., Volume 42 (2011), pp. 461-485 | DOI | Zbl

[25] Jean Dolbeault; Bruno Nazaret; Giuseppe Savaré On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., Volume 6 (2008) no. 2, pp. 477-494 | DOI | Zbl

[26] Ugo Gianazza; Giuseppe Savaré; Giuseppe Toscani The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 1, pp. 133-220 | DOI | MR | Zbl

[27] Basilis Gidas; Joel Spruck Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., Volume 34 (1981) no. 4, pp. 525-598 | DOI | MR | Zbl

[28] Pierre Grisvard Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman Publishing (Advanced Publishing Program), Boston, MA, 1985, xiv+410 pages | MR | Zbl

[29] Leonard Gross Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | DOI | MR | Zbl

[30] Guangyue Huang; Wenyi Chen Uniqueness for the solution of semi-linear elliptic Neumann problems in $ℝ^{3}$ , Commun. Pure Appl. Anal., Volume 7 (2008) no. 5, pp. 1269-1273 | DOI | MR | Zbl

[31] Joseph B. Keller Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation, J. Math. Phys., Volume 2 (1961), pp. 262-266 | DOI | MR | Zbl

[32] Rafał Latała; Krzysztof Oleszkiewicz Between Sobolev and Poincaré, Geometric aspects of functional analysis (Lecture Notes in Mathematics), Volume 1745, Springer, Berlin, 2000, pp. 147-168 | MR | Zbl

[33] Jean René Licois; Laurent Véron Un théorème d’annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci., Paris, Sér. I, Volume 320 (1995) no. 11, pp. 1337-1342 | MR | Zbl

[34] Jean René Licois; Laurent Véron A class of nonlinear conservative elliptic equations in cylinders, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 26 (1998) no. 2, pp. 249-283 | MR | Zbl

[35] Elliott H. Lieb; Walter E. Thirring Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in mathematical physics. Essays in honor of Valentine Bargmann, Princeton University Press, 1976, pp. 269-303 | Zbl

[36] Chang-Shou Lin; Wei-Ming Ni On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations (Trento, 1986) (Lecture Notes in Mathematics), Volume 1340, Springer, Berlin, 1988, pp. 160-174 | DOI | MR | Zbl

[37] Chang-Shou Lin; Wei-Ming Ni; Izumi Takagi Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, Volume 72 (1988) no. 1, pp. 1-27 | DOI | MR | Zbl

[38] Yasuhito Miyamoto Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal., Volume 256 (2009) no. 3, pp. 747-776 | DOI | MR | Zbl

[39] Yasuhito Miyamoto Global branch from the second eigenvalue for a semilinear Neumann problem in a ball, J. Differ. Equations, Volume 249 (2010) no. 8, pp. 1853-1870 http/www.sciencedirect.com/science/article/pii/S0022039610002469 | DOI | Zbl

[40] Wei-Ming Ni Diffusion, cross-diffusion, and their spike-layer steady states, Notices Am. Math. Soc., Volume 45 (1998) no. 1, pp. 9-18 | MR | Zbl

[41] Wei-Ming Ni; Izumi Takagi On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Am. Math. Soc., Volume 297 (1986) no. 1, pp. 351-368 | DOI | MR | Zbl

[42] Patrizia Pucci; James Serrin; Henghui Zou A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl., Volume 78 (1999) no. 8, pp. 769-789 | DOI | MR | Zbl

[43] Renate Schaaf Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, 1458, Springer, 1990, xx+141 pages | MR | Zbl

[44] Junping Shi Semilinear Neumann boundary value problems on a rectangle, Trans. Am. Math. Soc., Volume 354 (2002) no. 8, p. 3117-3154 (electronic) | DOI | MR | Zbl

[45] Joel A. Smoller; Arthur G. Wasserman Global bifurcation of steady-state solutions, J. Differ. Equations, Volume 39 (1981) no. 2, pp. 269-290 | DOI | MR | Zbl

[46] Cédric Villani Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009, xxii+973 pages (Old and new) | DOI | MR | Zbl

[47] Liping Wang; Juncheng Wei; Shusen Yan On Lin-Ni’s conjecture in convex domains, Proc. Lond. Math. Soc., Volume 102 (2011) no. 6, pp. 1099-1126 | DOI | MR | Zbl

[48] Juncheng Wei; Xingwang Xu Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in $ℝ^{3}$ , Pac. J. Math., Volume 221 (2005) no. 1, pp. 159-165 | DOI | MR | Zbl

[49] Chunshan Zhao Asymptotic behaviors of a class of $N$ -Laplacian Neumann problems with large diffusion, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 69 (2008) no. 8, pp. 2496-2524 | DOI | MR | Zbl

[50] Meijun Zhu Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differ. Equations, Volume 154 (1999) no. 2, pp. 284-317 | DOI | MR | Zbl

Cité par Sources :