Plateau borders in soap films and Gauss' capillarity theory

TL;DR

This paper rigorously derives the equilibrium law for Plateau borders in soap films within Gauss' capillarity theory, using a measure-theoretic approach to analyze the homotopic spanning condition and establish regularity of energy minimizers.

Contribution

It introduces a measure-theoretic reformulation of the homotopic spanning condition, enabling compactness and regularity results for Plateau borders in wet soap films and foams.

Findings

Derived the equilibrium law for Plateau borders in soap films.

Established regularity properties of energy minimizers.

Provided a measure-theoretic framework for homotopic spanning conditions.

Abstract

We provide, in the setting of Gauss' capillarity theory, a rigorous derivation of the equilibrium law for the three dimensional structures known as Plateau borders which arise in "wet" soap films and foams. A key step in our analysis is a complete measure-theoretic overhaul of the homotopic spanning condition introduced by Harrison and Pugh in the study of Plateau's laws for two-dimensional area minimizing surfaces ("dry" soap films). This new point of view allows us to obtain effective compactness theorems and energy representation formulae for the homotopic spanning relaxation of Gauss' capillarity theory which, in turn, lead to prove sharp regularity properties of energy minimizers. The equilibrium law for Plateau borders in wet foams is also addressed as a (simpler) variant of the theory for wet soap films.