An existence theorem for sliding minimal sets

We prove an existence theorem for the sliding boundary variant of the Plateau problem for $2$-dimensional sets in ${\mathbb{R}}^n$. The simplest case of sufficient condition is when $n=3$ and the boundary $\mathrm{\Gamma }$ is a finite disjoint union of smooth closed curves contained in the boundary of a convex body, but the main point of our sufficient condition is to prevent the limits in measure of a minimizing sequence to have singularities of type $\mathbb{Y}$ along $\mathrm{\Gamma }$.