Abstract:
We link Brakke's "soap films" covering construction with the theory of finite perimeter
sets, in order to study Plateau's problem without fixing a priori
the topology of the solution. The minimization is set up in the class of $BV$ functions
defined on a double covering space of the complement of an $(n − 2)$-dimensional
smooth compact manifold $S$ without boundary. The main novelty
of our approach stands in the presence of a suitable constraint on the fibers, which
couples together the covering sheets. The model allows to avoid all issues
concerning the presence of the boundary $S$. The constraint is lifted in a natural way
to Sobolev spaces, allowing also an approach based on $Γ$-convergence theory.
