A non-parametric Plateau problem with partial free boundary

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 t...

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Bibliographic Details
Published inarXiv.org
Main Authors Bellettini, Giovanni, Marziani, Roberta, Scala, Riccardo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.01.2022
Subjects
Dirichlet problem
Free boundaries
Minimal surfaces
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Summary: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 \(\partial \Omega\).
ISSN:2331-8422