The Plateau problem

In its classical formulation Plateau problem asks to find the surface of minimal area spanning a given boundary. Understanding the proper notion of surface, area and of “spanning a  boundary” has lead since the pioneering works of Almegren-De Giorgi/Federe-Fleming/Reifenberg in the 50’s to the development of beautiful and powerful tools in Geometric Measure Theory.

I will review the state of the art on the Plateau problem by presenting a series of old and new results concerning existence and regularity of solutions.