Minimal surfaces, namely surfaces with everywhere vanishing mean curvature, are the classical model for soap films spanning a given boundary wire. A major issue with this model is the complete lack of a length scale capturing the mechanical properties of soap films due to surface tension. In this talk, I will discuss how to overcome this issue by modeling soap films as "thick" sets of finite perimeter enclosing a small volume and satisfying a suitable spanning condition. The resulting variational problem is a singular capillarity problem. I will discuss existence and regularity of solutions, as well as the possibility to recover the classical Plateau's problem in the limit of singular capillarity approximations when the enclosed volume approaches zero.

Based on joint works with Darren King (UT Austin), Francesco Maggi (UT Austin), and Antonello Scardicchio (ICTP Trieste).