A non-parametric Plateau problem with partial free boundary

TL;DR

This paper studies a non-parametric Plateau problem with partial boundary conditions, allowing free contact with a plane, establishing existence, regularity, and comparing solutions with classical minimal surfaces.

Contribution

It introduces a new non-parametric Plateau problem with partial boundary data and free boundary conditions, proving existence, regularity, and equivalence with classical solutions in specific cases.

Findings

Existence of solutions with partial boundary data and free contact.

Proved regularity of the minimal surface solutions.

Established equivalence with classical minimal surfaces when boundary data is on at most two arcs.

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