A non-parametric Plateau problem with partial free boundary

Abstract

We consider a Plateau problem in codimension-one in the non-parametric setting. A Dirichlet boundary datum is given only on a part of the boundary of a convex domain in the plane. Where the Dirichlet datum is not prescribed, we allow the solution to have a free contact with the plane domain. We then show existence of a solution, and prove some regularity for the corresponding area-minimizing surface. Finally we compare our solutions to the solutions provided by Meeks & Yau for the classical Plateau problem. In particular we show that they are equivalent, at least in the case where the Dirichlet boundary datum is assigned in at most 2 connected components of the boundary of the domain. This result was obtained in collaboration with Giovanni Bellettini & Riccardo Scala.

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