A non-parametric Plateau problem with partial free boundary

Bellettini, Giovanni; Marziani, Roberta; Scala, Riccardo

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1035-1098

We consider a Plateau problem in codimension $1$ in the non-parametric setting, where a Dirichlet boundary datum is assigned only on part of the boundary $\partial \Omega$ of a bounded convex domain $\Omega \subset {ℝ}^2$. Where the Dirichlet datum is not prescribed, we allow a free contact with the horizontal plane. We show existence of a solution, and prove regularity for the corresponding area-minimizing surface. We compare these solutions with the classical minimal surfaces of Meeks and Yau, and show that they are equivalent when the Dirichlet boundary datum is assigned on at most $2$ disjoint arcs of $\partial \Omega$.

Nous considérons un problème de Plateau en codimension $1$ dans un cadre non paramétrique, où une donnée de Dirichlet n’est assignée que sur une partie de la frontière $\partial \Omega$ d’un domaine convexe borné $\Omega \subset {ℝ}^2$. Là où la donnée de Dirichlet n’est pas prescrite, nous autorisons un contact libre avec le plan horizontal. Nous montrons l’existence d’une solution, et prouvons la régularité de la surface minimale correspondante. Nous comparons ces solutions avec les surfaces minimales classiques de Meeks et Yau, et montrons qu’elles sont équivalentes lorsque la donnée de Dirichlet est assignée sur au plus $2$ arcs disjoints de $\partial \Omega$.

Classification : 49J45, 49Q05, 49Q15, 28A75
Keywords: Plateau problem, area functional, minimal surfaces, relaxation, Cartesian currents
Mots-clés : Problème de Plateau, fonctionnelle d’aire, surfaces minimales, relaxation, courants cartésiens

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