A non-parametric Plateau problem with partial free boundary

Giovanni Bellettini; Roberta Marziani; Riccardo Scala

doi:10.5802/jep.273

A non-parametric Plateau problem with partial free boundary

Giovanni Bellettini ¹; Roberta Marziani ²; Riccardo Scala ²

¹Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy, & International Centre for Theoretical Physics ICTP, Mathematics Section, 34151 Trieste, Italy
²Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy

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

Abstract (Original language)
Résumé

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 $\partial Ω$ of a bounded convex domain $Ω \subset ℝ^{2}$ . 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 $\partial Ω$ .

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 $\partial Ω$ d’un domaine convexe borné $Ω \subset ℝ^{2}$ . 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 $\partial Ω$ .

Article information
Cite this article

Received: 2023-09-18
Accepted: 2024-09-18
Published online: 2024-10-08

MR Zbl

DOI: 10.5802/jep.273

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

Author's affiliations:

Giovanni Bellettini ¹ ; Roberta Marziani ² ; Riccardo Scala ²

¹ Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy, & International Centre for Theoretical Physics ICTP, Mathematics Section, 34151 Trieste, Italy
² Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Giovanni Bellettini; Roberta Marziani; Riccardo Scala. A non-parametric Plateau problem with partial free boundary. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 1035-1098. doi: 10.5802/jep.273

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