Log-concavity and anti-maximum principles for semilinear and linear elliptic equations

François Hamel; Nikolai Nadirashvili

doi:10.5802/crmath.810

Article de recherche - Equations aux dérivées partielles

Log-concavity and anti-maximum principles for semilinear and linear elliptic equations
[Log-concavité et principes d’anti-maximum pour des équations elliptiques linéaires et semi-linéaires]

François Hamel ¹ ; Nikolai Nadirashvili ¹

¹ Aix Marseille Univ, CNRS, I2M, Marseille, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1555-1574

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par Henri Berestycki et al..

Abstract (VO)
Résumé

This paper is concerned with existence and qualitative properties of positive solutions of semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We show the existence of positive solutions in the vicinity of the linear equation and the log-concavity of the solutions when the domain is strictly convex. We also review the standard results on the log-concavity or the more general quasi-concavity of solutions of elliptic equations. The existence and other convergence results especially rely on the maximum principle, on a quantified version of the anti-maximum principle, on the Schauder fixed point theorem, and on some a priori estimates.

Cet article porte sur des questions d’existence et des propriétés qualitatives de solutions strictement positives d’équations elliptiques semi-linéaires dans des domaines bornés avec des conditions au bord de type Dirichlet. Nous montrons l’existence de solutions strictement positives au voisinage d’une équation linéaire et la log-concavité des solutions lorsque le domaine est strictement convexe. Nous présentons également une synthèse des résultats classiques sur la log-concavité et sur la notion plus générale de quasi-concavité pour des équations elliptiques. L’existence de solutions et des résultats supplémentaires de convergence reposent notamment sur le principe du maximum, sur une version quantifiée du principe d’anti-maximum, sur le théorème de point fixe de Schauder, et sur des estimations a priori.

Reçu le : 2025-06-15
Révisé le : 2025-11-24
Accepté le : 2025-11-24
Publié le : 2025-12-12

DOI : 10.5802/crmath.810

Classification : 35A16, 35B20, 35B30, 35B50, 35J15, 35J61
Keywords: Semilinear elliptic equations, qualitative properties, log-concavity, anti-maximum principle
Mots-clés : Équations elliptiques semi-linéaires, propriétés qualitatives, log-concavité, principe d’anti-maximum

Affiliations des auteurs :

François Hamel ¹ ; Nikolai Nadirashvili ¹

¹ Aix Marseille Univ, CNRS, I2M, Marseille, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fran\c{c}ois Hamel and Nikolai Nadirashvili},
     title = {Log-concavity and anti-maximum principles for semilinear and linear elliptic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1555--1574},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.810},
     language = {en},
}

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François Hamel; Nikolai Nadirashvili. Log-concavity and anti-maximum principles for semilinear and linear elliptic equations. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1555-1574. doi: 10.5802/crmath.810

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