Smoothness of collapsed regions in a capillarity model for soap films

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.

arXiv comment33 pages, 8 figures

References in corpus (5)

• The Singular Structure and Regularity of Stationary and Minimizing Varifolds
• Soap films with gravity and almost-minimal surfaces
• Collapsing and the convex hull property in a soap film capillarity model
• Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds
• The singular set of minimal surfaces near polyhedral cones

Cited by in corpus (6)

• Collapsing and the convex hull property in a soap film capillarity model
• Plateau's problem as a singular limit of capillarity problems
• On the relaxation of Gauss's capillarity theory under spanning conditions
• Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited
• Classical solutions to the soap film capillarity problem for plane boundaries
• Regularity for Minimizers of a Planar Partitioning Problem with Cusps