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

TL;DR

This paper proves $C^{1,eta}$-regularity at the boundary for two-dimensional sliding almost minimal sets in three-dimensional space, advancing understanding of boundary regularity in geometric measure theory.

Contribution

It establishes boundary regularity results for sliding almost minimal sets, potentially aiding solutions to the Plateau problem with sliding boundary conditions.

Findings

Boundary regularity established for sliding almost minimal sets.

Potential existence results for Plateau problem with sliding boundaries.

Advances in boundary regularity theory in geometric measure theory.

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.