Smoothness of collapsed regions in a capillarity model for soap films

TL;DR

This paper proves that in a capillarity model for soap films, collapsed regions are smooth outside small singular sets and do not contain certain complex singularities, providing insight into the structure of limiting Plateau surfaces.

Contribution

It establishes the smoothness of collapsed regions in a capillarity model and links the absence of certain singularities to the behavior of approximating minimizers.

Findings

Collapsed regions are smooth outside small singular sets.

Collapsed regions cannot have Y-points or T-points.

Supports that singularities in Plateau surfaces are 'wetted' by minimizers.

Abstract

We study generalized minimizers in the soap film capillarity model introduced in [arXiv:1807.05200,arXiv:1907.00551]. Collapsed regions of generalized minimizers are shown to be smooth outside of dimensionally small singular sets, which are thus empty in physical dimensions. Since generalized minimizers converge to Plateau's type surfaces in the vanishing volume limit, the fact that collapsed regions cannot exhibit $Y$-points and $T$-points (which are possibly present in the limit Plateau's surfaces) gives the first strong indication that singularities of the limit Plateau's surfaces should always be "wetted" by the bulky regions of the approximating generalized minimizers.