Plateau's Problem as a Singular Limit of Capillarity Problems

Abstract

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection principle for Plateau's problem, points at physical features of soap films that are unaccessible by simply looking at minimal surfaces, and opens several challenging questions.

Regularity of almost-minimizers of Hölder-coefficient surface energies

“…In this appendix we provide a proof for the change of variable formula for sets of locally finite perimeter with bounded, continuous integrands depending on both x and ν E . This is a generalization of [29, Proposition 17.1] and [27, Theorem A.1].…”

“…The analogous physical phenomenon occurs in soap films as they seek to minimize surface tension, an equivalent to minimizing surface area. The existence and regularity of solutions to the Plateau problem has been the subject of study in a variety of settings and continues to be a centerpiece of much mathematical research (to name a few, see [20,33,9,34,1,44,25,12,27]). A natural generalization of the Plateau problem is to study minimizers of surface energies other than surface area.…”

Plateau's Problem as a Singular Limit of Capillarity Problems

“…Math. in 2020 with DOI 10.1002/cpa.21951; see [39]. The revision does not contain any major changes with respect to the previous version: in particular, all main results stated in [39] remain valid.…”

A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities

“…The energetics of soap films are governed by surface tension and its static properties by its total surface area and the conformation of the bounding frame. Neglecting contributions from finite thickness [22] and external forces such as gravity [23], the energy of a soap film Σ is simply proportional to its total surface area E=σ∫italicΣ dA, where σ is the surface tension (we also neglect elasticity of the boundary wire [24]). Critical points of the area functional are minimal surfaces and characterize the soap film morphology as a surface with vanishing mean curvature.…”