The goal of this work is to use a phase field method to approximate the notorious Plateau problem. To this aim, we want to generalise the functional, introduced by M. Bonnivard, A. Lemenant and F. Santambrogio for the Steiner problem, to the Plateau problem with a Reifenberg formulation. The novelty of this approach is to deal with a topological constrain by penalizing some geodesic distance, which must be defined.

We first properly define the Plateau problem we consider, and give some regularity results, then we present a Gamma-convergence type result for the new approximation functional. Finally, this analysis allows us to obtain some numerical simulations of the minimizers for the Plateau problem.