The incompatibility between Plateau's laws and stable solutions to the Allen--Cahn equation is resolved by the formulation and analysis of a new model for soap films as small volume regions with diffused interfaces. As a result, Plateau-type singularities are approximated by stable solutions to free boundary problems for modified Allen--Cahn equations. Underlying our approach is the study of a hierarchy of Plateau problems that showcases the newly introduced diffused interface model at the top, a soap film capillarity model with sharp interfaces and bulk spanning at the intermediate level, and the classical Plateau model at the bottom. Central to our analysis is a measure-theoretic revision of the topological notion of homotopic spanning that has been behind much recent progress on the classical Plateau problem.

This is joint work with Michael Novack (CMU Pittsburgh) and Daniel Restrepo (Johns Hopkins University).