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

Enache, Cristian

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

Dans cette note, nous obtenons un principe du maximum pour une combinaison fonctionnelle appropriée de $u(\mathbf{x})$ et ${|\mathrm{\nabla }u|}^2$, où $u(\mathbf{x})$ 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 ${\mathbb{R}}^{N+1}$.

In this note we derive a maximum principle for an appropriate functional combination of $u(\mathbf{x})$ and ${|\mathrm{\nabla }u|}^2$, where $u(\mathbf{x})$ 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 ${\mathbb{R}}^{N+1}$.

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