Soap films in equilibrium hanging from a boundary wire are classically modeled as two-dimensional surfaces with everywhere vanishing mean curvature, thus motivating the study of minimal surfaces. By neglecting their intrinsic three-dimensional structure, this model fails at capturing some of the physical properties of soap films observed in the experiments. In this talk, I will present a variational model, based on Gauss' theory of capillarity, which aims at overcoming this issue by regarding soap films as ``thick?? sets of finite perimeter enclosing a given volume of fluid and spanning the prescribed boundary wire. I will discuss the corresponding existence theory, the geometry and the regularity properties of the minimizers, and the asymptotic behavior in the vanishing volume limit.
Based on joint works with Darren King and Francesco Maggi (UT Austin), and Antonello Scardicchio (ICTP Trieste).