2019/06/20 by Vicent Gimeno, V. Gimeno, Steen Markvorsen +6
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Analysis Techniques #Applied Physics (physics.app-ph) #Benchmark (surveying) #Bounded function #Combinatorics #Computer science #Differential Geometry (math.DG) #Divergence (linguistics) #FOS: Mathematics #FOS: Physical sciences #Function (biology) #Geology #Geometric Analysis and Curvature Flows #Geometry #Graph #Gravitational singularity #Mathematical analysis #Mathematics #Minimal surface #Plateau (mathematics) #Point processes and geometric inequalities #Set (abstract data type) #Space (punctuation) #Surface (topology) #Upper and lower bounds #Vertex (graph theory) #math.DG #physics.app-ph
paper · pdf · doi:10.48550/arxiv.1906.08537
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2019/06/20 · arxiv created 2019/07/19 · arxiv updated 2019/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Real foams can be viewed as a geometrically well-organized dispersion of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at the Plateau borders according to the celebrated Plateau's laws. In this paper we estimate from below the surface area of a spherically bounded piece of a foam. Our main tool is a new version of the divergence theorem which is adapted to the specific geometry of a foam with special attention to its classical Plateau singularities. As a benchmark application of our results we obtain lower bounds for the fundamental cell of a Kelvin foam, lower bounds for the so-called cost function, and for the difference of the pressures appearing in minimal periodic foams. Moreover, we provide an algorithm whose input is a set of isolated points in space and whose output is the best lower bound estimate for the area of a foam that contains the given set as its vertex set.