Minimal Graphs in $\mathbb{H}^n\times \mathbb{R}$ and $\mathbb{R}^{n+1}$

[Graphes minimaux dans $\mathbb{H}^n\times \mathbb{R}$ et $\mathbb{R}^{n+1}$]

Sà Earp, Ricardo ; Toubiana, Eric

Annales de l'Institut Fourier, Tome 60 (2010), p. 2373-2402 / Harvested from Numdam

Résumé

Nous construisons des barrières géométriques dans $\mathbb{H}^n\times \mathbb{R}.$

Nous prouvons l’existence et l’unicité d’une solution de l’équation du graphe vertical minimal sur l’intérieur d’un polyhèdre convexe de $\mathbb{H}^n$ qui se prolonge sur l’intérieur de chaque face, prenant la valeur infinie sur une face et la valeur zéro sur les autres faces.

Dans $\mathbb{H}^n\times \mathbb{R}$, nous résolvons le problème de Dirichlet pour l’équation du graphe vertical minimal sur un domaine $C^0$ convexe $\Omega \subset \mathbb{H}^n$ prenant des données continues arbitraires sur le bord fini et le bord asymptotique de $\Omega $.

Nous prouvons l’existence d’une autre hypersurface de type Scherk, donnée par la solution de l’équation du graphe vertical minimal sur l’intérieur d’un certain polyhèdre admissible prenant alternativement les valeurs $+\infty $ et $-\infty $ sur les faces adjacentes.

Nous établissons des resultats analogues pour des graphes minimaux dans $\mathbb{R}^{n+1}$.

Abstract

We construct geometric barriers for minimal graphs in $\mathbb{H}^n\times \mathbb{R}.$

We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in $\mathbb{H}^n$ extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on the other faces.

In $\mathbb{H}^n\times \mathbb{R}$, we solve the Dirichlet problem for the vertical minimal equation in a $C^0$ convex domain $\Omega \subset \mathbb{H}^n$ taking arbitrarily continuous finite boundary and asymptotic boundary data.

We prove the existence of another Scherk type hypersurface, given by the solution of the vertical minimal equation in the interior of certain admissible polyhedron taking alternatively infinite values $+\infty $ and $-\infty $ on adjacent faces of this polyhedron.

We establish analogous results for minimal graphs when the ambient is the Euclidean space $\mathbb{R}^{n+1}$.

Détail

Publié le : 2010-01-01

MR 2849265 | Zbl 1225.53060

DOI : https://doi.org/10.5802/aif.2611
Classification:  53C42,  35J25
Mots clés: problème de Dirichlet, équation de surface minimale, graphe vertical, processus de Perron, barrière, domaine convexe, bord asymptotique, hypersurface de translation, hypersurface de Scherk

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