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Global Lipschitz and Sobolev estimates for the Monge–Ampère eigenfunctions of general bounded convex domains
[Estimations Lipschitz globales et de Sobolev pour les fonctions propres de l’opérateur de Monge-Ampère sur des domaines convexes bornés généraux]
Nam Q. Le1
1Indiana University, Bloomington, Department of Mathematics, 831 E 3rd St, Bloomington, IN 47405 (USA)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 467-487

We show that the Monge–Ampère eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge–Ampère equations of the form $\det D^2 u =M|u|^p$ with zero boundary condition on general bounded convex domains in $\mathbb{R}^n$ within the sharp threshold $p>n-2$. As a consequence, we obtain global $W^{2, 1}$ estimates for these solutions.

Nous montrons que les fonctions propres de l’opérateur de Monge–Ampère des domaines convexes bornés généraux sont globalement lipschitziennes. Le même résultat est valable pour les solutions convexes des équations de Monge–Ampère dégénérées de la forme $\det D^2 u =M|u|^p$ avec condition aux limites nulle sur les domaines convexes bornés généraux de $\mathbb{R}^n$ dans le seuil précis $p>n-2$. En conséquence, nous obtenons des estimations globales dans $W^{2, 1}$ pour ces solutions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1854
Classification : 35J96, 35J70, 35B45, 35B51
Keywords: Monge–Ampère eigenfunctions, degenerate Monge–Ampère equation, Lipschitz estimate, Sobolev estimate, subsolution, comparison principle

Nam Q. Le  1

1 Indiana University, Bloomington, Department of Mathematics, 831 E 3rd St, Bloomington, IN 47405 (USA)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Nam Q. Le. Global Lipschitz and Sobolev estimates for the Monge–Ampère eigenfunctions of general bounded convex domains. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 467-487. doi: 10.5802/afst.1854
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