Calculus of Variations and Geometric Measure Theory

M. Novack

On the relaxation of Gauss's capillarity theory under spanning conditions

created by novack on 10 Nov 2023
modified on 15 May 2024


Published Paper

Inserted: 10 nov 2023
Last Updated: 15 may 2024

Journal: Journal of Geometric Analysis
Year: 2024

ArXiv: 2311.05603v2 PDF


We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely "wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class.