Calculus of Variations and Geometric Measure Theory

G. Bellettini - R. Marziani - R. Scala

A non-parametric Plateau problem with partial free boundary

created by scala on 29 Oct 2021
modified by marziani on 17 Jan 2022


Submitted Paper

Inserted: 29 oct 2021
Last Updated: 17 jan 2022

Year: 2021


We consider a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on a part of the boundary $\partial \Omega$ of a convex domain $\Omega\subset\mathbb R^2$. Where the Dirichlet datum is not prescribed, we allow the solution to have a free contact with the plane domain. We show existence of a solution, and prove some regularity for the corresponding minimal surface. Finally we compare the solutions we find with classical solutions provided by Meeks and Yau, and show that they are equivalent at least in the case that the Dirichlet boundary datum is assigned in at most $2$ connected components of $\partial \Omega$.