logo AFST
Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates
Jean Dolbeault; Michał Kowalczyk
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, p. 949-977

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.

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.

Published online : 2017-12-13
DOI : https://doi.org/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(ρ,N) condition, bifurcation, multiplicity, generalized entropy methods, heat flow, nonlinear diffusion, spectral gap inequality, Poincaré inequality, improved inequalities, non-Lipschitz nonlinearity, compact support principle
@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {4},
     year = {2017},
     pages = {949-977},
     doi = {10.5802/afst.1557},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_4_949_0}
}
Dolbeault, Jean; Kowalczyk, Michał. Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 949-977. doi : 10.5802/afst.1557. afst.centre-mersenne.org/item/AFST_2017_6_26_4_949_0/

[1] Anton Arnold; Jean-Philippe Bartier; Jean Dolbeault Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., Tome 5 (2007) no. 4, pp. 971-979 | Article | Zbl 1146.60063

[2] Dominique Bakry; Michel Émery Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci., Paris, Sér. I, Tome 299 (1984) no. 15, pp. 775-778 | MR MR772092 (86f:60097) | Zbl 0563.60068

[3] Dominique Bakry; Michel Émery Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci., Paris, Sér. I, Tome 301 (1985) no. 8, pp. 411-413 | MR 808640 (86k:60141) | Zbl 0579.60079

[4] Dominique Bakry; Michel Ledoux Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J., Tome 85 (1996) no. 1, pp. 253-270 | Article | MR 1412446 (97h:53034) | Zbl 0870.60071

[5] Mikhaël Balabane; Jean Dolbeault; Hichem Ounaies Nodal solutions for a sublinear elliptic equation, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Tome 52 (2003) no. 1, pp. 219-237 | Article | MR 1938658 (2003h:35062) | Zbl 1087.35033

[6] William Beckner A generalized Poincaré inequality for Gaussian measures, Proc. Am. Math. Soc., Tome 105 (1989) no. 2, pp. 397-400 | MR MR954373 (89m:42027) | Zbl 0677.42020

[7] Marie-Françoise Bidaut-Véron; Laurent Véron Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., Tome 106 (1991) no. 3, pp. 489-539 (erratum in ibid. 112 (1993), no. 2, p. 445) | Article | MR 1134481 (93a:35045) | Zbl 0755.35036

[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, Tome 21 (1996) no. 3-4, pp. 507-520 | Article | MR 1387457 (97d:35053) | Zbl 0854.35033

[9] Carmen Cortázar; Manuel Elgueta; Patricio Felmer Uniqueness of positive solutions of Δu+f(u)=0 in N ,N3, Arch. Ration. Mech. Anal., Tome 142 (1998) no. 2, pp. 127-141 | Article | MR 1629650 (2000b:35059) | Zbl 0912.35059

[10] Jérôme Demange Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie (2005) (Ph. D. Thesis)

[11] Jérôme Demange Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., Tome 254 (2008) no. 3, pp. 593-611 | Article | MR 2381156 (2009e:58037) | Zbl 1133.58012

[12] Jean Dolbeault; Maria J. Esteban A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities, J. Numer. Math., Tome 20 (2013) no. 3-4, p. 233--249 | Zbl 1267.65075

[13] Jean Dolbeault; Maria J. Esteban Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, Tome 27 (2014) no. 3, pp. 435-465 http://stacks.iop.org/0951-7715/27/i=3/a=435 | Article | Zbl 1292.35033

[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, Tome 34 (2013) no. 1, pp. 99-112 | Article | Zbl 1263.26029

[15] Jean Dolbeault; Maria J. Esteban; Michal Kowalczyk; Michael Loss Improved interpolation inequalities on the sphere, Discrete Contin. Dyn. Syst., Ser. S, Tome 7 (2014) no. 4, pp. 695-724 | Article | Zbl 1290.26022

[16] Jean Dolbeault; Maria J. Esteban; Ari Laptev Spectral estimates on the sphere, Anal. PDE, Tome 7 (2014) no. 2, pp. 435-460 | Article | Zbl 1293.35183

[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, Tome 351 (2013) no. 11–12, pp. 437 -440 | Article | Zbl 1276.58007

[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., Tome 90 (2014) no. 2, pp. 525-550 | Article | Zbl 1320.26017

[19] Jean Dolbeault; Maria J. Esteban; Michael Loss Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., Tome 267 (2014) no. 5, pp. 1338 -1363 | Article | Zbl 1294.58005

[20] Jean Dolbeault; Maria J. Esteban; Michael Loss Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders, C. R., Math., Acad. Sci. Paris, Tome 353 (2015) no. 9, pp. 813-818 | Article | MR 3377678 | Zbl 1323.35035

[21] Jean Dolbeault; Maria J. Esteban; Michael Loss Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., Tome 206 (2016) no. 2, pp. 397-440 | Article | MR 3570296 | Zbl 06664755

[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, Tome 26 (2017) no. 2, pp. 351-379 | Article | Zbl 06754494

[24] Jean Dolbeault; Maria J. Esteban; Gabriella Tarantello; Achilles Tertikas Radial symmetry and symmetry breaking for some interpolation inequalities, Calc. Var. Partial Differ. Equ., Tome 42 (2011), pp. 461-485 | Article | Zbl 1246.26014

[25] Jean Dolbeault; Bruno Nazaret; Giuseppe Savaré On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., Tome 6 (2008) no. 2, pp. 477-494 | Article | Zbl 1149.35330

[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., Tome 194 (2009) no. 1, pp. 133-220 | Article | MR 2533926 (2010i:35371) | Zbl 1223.35264

[27] Basilis Gidas; Joel Spruck Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., Tome 34 (1981) no. 4, pp. 525-598 | Article | MR 615628 (83f:35045) | Zbl 0465.35003

[28] Pierre Grisvard Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, Tome 24, Pitman Publishing (Advanced Publishing Program), Boston, MA, 1985, xiv+410 pages | MR MR775683 (86m:35044) | Zbl 0695.35060

[29] Leonard Gross Logarithmic Sobolev inequalities, Am. J. Math., Tome 97 (1975) no. 4, pp. 1061-1083 | Article | MR 54 #8263 | Zbl 0318.46049

[30] Guangyue Huang; Wenyi Chen Uniqueness for the solution of semi-linear elliptic Neumann problems in 3 , Commun. Pure Appl. Anal., Tome 7 (2008) no. 5, pp. 1269-1273 | Article | MR 2410880 (2009f:35090) | Zbl 1144.35380

[31] Joseph B. Keller Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation, J. Math. Phys., Tome 2 (1961), pp. 262-266 | Article | MR 0121101 (22 #11847) | Zbl 0099.06901

[32] Rafał Latała; Krzysztof Oleszkiewicz Between Sobolev and Poincaré, Geometric aspects of functional analysis (Lecture Notes in Mathematics) Tome 1745, Springer, Berlin, 2000, pp. 147-168 | MR MR1796718 (2002b:60025) | Zbl 0986.60017

[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, Tome 320 (1995) no. 11, pp. 1337-1342 | MR 1338283 (96e:58166) | Zbl 0839.53031

[34] Jean René Licois; Laurent Véron A class of nonlinear conservative elliptic equations in cylinders, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Tome 26 (1998) no. 2, pp. 249-283 | MR 1631581 (99g:35038) | Zbl 0918.35051

[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 0342.35044

[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) Tome 1340, Springer, Berlin, 1988, pp. 160-174 | Article | MR 974610 (90d:35101) | Zbl 0704.35050

[37] Chang-Shou Lin; Wei-Ming Ni; Izumi Takagi Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, Tome 72 (1988) no. 1, pp. 1-27 | Article | MR 929196 (89e:35075) | Zbl 0676.35030

[38] Yasuhito Miyamoto Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal., Tome 256 (2009) no. 3, pp. 747-776 | Article | MR 2484935 (2010d:35130) | Zbl 1166.35024

[39] Yasuhito Miyamoto Global branch from the second eigenvalue for a semilinear Neumann problem in a ball, J. Differ. Equations, Tome 249 (2010) no. 8, pp. 1853-1870 http/www.sciencedirect.com/science/article/pii/S0022039610002469 | Article | Zbl 1201.35100

[40] Wei-Ming Ni Diffusion, cross-diffusion, and their spike-layer steady states, Notices Am. Math. Soc., Tome 45 (1998) no. 1, pp. 9-18 | MR 1490535 (99a:35132) | Zbl 0917.35047

[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., Tome 297 (1986) no. 1, pp. 351-368 | Article | MR 849484 (87k:35091) | Zbl 0635.35031

[42] Patrizia Pucci; James Serrin; Henghui Zou A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl., Tome 78 (1999) no. 8, pp. 769-789 | Article | MR 1715341 (2001j:35095) | Zbl 0952.35045

[43] Renate Schaaf Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, Tome 1458, Springer, 1990, xx+141 pages | MR 1090827 (92a:34003) | Zbl 0780.34010

[44] Junping Shi Semilinear Neumann boundary value problems on a rectangle, Trans. Am. Math. Soc., Tome 354 (2002) no. 8, p. 3117-3154 (electronic) | Article | MR 1897394 (2003b:35083) | Zbl 0992.35031

[45] Joel A. Smoller; Arthur G. Wasserman Global bifurcation of steady-state solutions, J. Differ. Equations, Tome 39 (1981) no. 2, pp. 269-290 | Article | MR 607786 (82d:58056) | Zbl 0425.34028

[46] Cédric Villani Optimal transport, Grundlehren der Mathematischen Wissenschaften, Tome 338, Springer, 2009, xxii+973 pages (Old and new) | Article | MR 2459454 | Zbl 1156.53003

[47] Liping Wang; Juncheng Wei; Shusen Yan On Lin-Ni’s conjecture in convex domains, Proc. Lond. Math. Soc., Tome 102 (2011) no. 6, pp. 1099-1126 | Article | MR 2806101 (2012f:35191) | Zbl 1236.35077

[48] Juncheng Wei; Xingwang Xu Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in 3 , Pac. J. Math., Tome 221 (2005) no. 1, pp. 159-165 | Article | MR 2194150 (2006k:35106) | Zbl 1144.35382

[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, Tome 69 (2008) no. 8, pp. 2496-2524 | Article | MR 2446347 (2009k:35093) | Zbl 1153.35042

[50] Meijun Zhu Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differ. Equations, Tome 154 (1999) no. 2, pp. 284-317 | Article | MR 1691074 (2000c:35078) | Zbl 0927.35041