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

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 apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold.