Mathematics

The most famous context where minimal surfaces arise is the Plateau Problem: one tries to minimize the area of a surface  whose boundary is a fixed closed curve .If we want to solve slightly different problems, where the solution is not globally area-minimizing and/or with different boundary conditions, minimizing methods might not work or might provide just trivial solutions. In that cases it can be convenient to obtain the solution of a PDE as a stationary point for a suitable functional. The min-max method is a way to prove the existence of such stationary solutions.In this talk I will give an overview about these issues; then I will explain the basic ideas of min-max theory applied to the problem of finding a minimal surface  in a container  with a fixed angle condition at the boundary. This part is based on a joint work with Guido De Philippis.