A non-parametric Plateau problem with partial free boundary

2022/01/16 by Giovanni Bellettini, Bellettini, Giovanni, Roberta Marziani +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2201.06145

openalex publication_date 2022/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a Plateau problem in codimension 1 in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary ∂ Ω of a bounded convex domain Ω⊂ℝ². 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 minimal surface. Finally 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 in at most 2 disjoint arcs of ∂ Ω.

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