A direct approach to Plateau's problem

I will recall what the Plateau's problem is and summarize the state of the art. I will present a compactness principle valid under fairly general assumptions on the class of competitors, which can be applied to solve different formulations of the Plateau's problem. The approach is the minimization of the d-Hausdorff measure among a family of d-rectifiable closed subsets of $R^n$. Such class is then specified to give meaning to boundary conditions. The obtained minimizers are regular up to a set of dimension less than $d-1$. I will also present a variant of this compactness principle for the minimization of more general energies, generated by elliptic integrands (in the sense of Almgren).