Local $C^{1,\beta }$-regularity at the boundary of two dimensional sliding almost minimal sets in ${\mathbb{R}}^3$

Abstract

In this paper, we will give a $C^{1,\beta }$-regularity result on the boundary for two dimensional sliding almost minimal sets in ${\mathbb{R}}^3$. This effect may lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in \cite{David:2014p} in the case that the boundary is a 2-dimensional smooth manifold.