The Plateau problem in the Calculus of Variations

This is a survey paper written for a course held for the Ph. D. program in Pure and Applied Mathematics at Politecnico di Torino during autumn 2018. The course has been dedicated to an overview of the main techniques for solving the Plateau problem, that is to find a surface with minimal area that spans a given boundary curve in the space. This problem dates back to the physical experiments of Plateau who tried to understand the possible configurations of soap films. From the mathematical point of view, the problem is very hard and a lot of possible formulations are available: perhaps still today none of these answers is the answer to the original formulation by Plateau. In this paper, first of all we will briefly introduce the problem showing that, at least in the smooth case, if the first variation of the area vanishes then the surface must have zero mean curvature. Then we will describe how the classical solution by Douglas and Radó works, and we will pass to modern formulations of the problem in the context of Geometric Measure Theory: sets of finite perimeter, currents, and minimal sets.