Inserted: 29 oct 2021
Last Updated: 17 jan 2022
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 $\partial \Omega$.