Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

Cristian Enache

doi:10.1016/j.crma.2013.10.035

Partial differential equations/Differential geometry

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems
[Principes du maximum et inégalités isopérimétriques pour certains problèmes du type Monge–Ampère]

Présenté par : Haïm Brezis

Cristian Enache ¹

¹ Research group of the project PN-II-ID-PCE-2012-4-0021, “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 37-42.

Résumé
Abstract

Dans cette note, nous obtenons un principe du maximum pour une combinaison fonctionnelle appropriée de $u (x)$ et ${| \nabla u |}^{2}$ , où $u (x)$ est une solution classique strictement convexe à une classe générale dʼéquations du type Monge–Ampère. Ce principe du maximum est ensuite utilisé pour établir certaines inégalités isopérimétriques dʼintérêt dans la théorie de surfaces de courbure de Gauss constante dans $R^{N + 1}$ .

In this note we derive a maximum principle for an appropriate functional combination of $u (x)$ and ${| \nabla u |}^{2}$ , where $u (x)$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in $R^{N + 1}$ .

Reçu le : 2013-10-08
Accepté le : 2013-10-30
Publié le : 2013-12-23

DOI : 10.1016/j.crma.2013.10.035

Affiliations des auteurs :

Cristian Enache ¹

¹ Research group of the project PN-II-ID-PCE-2012-4-0021, “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

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     title = {Maximum principles and isoperimetric inequalities for some {Monge{\textendash}Amp\`ere-type} problems},
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Cristian Enache. Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems. Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 37-42. doi : 10.1016/j.crma.2013.10.035. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.035/

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