A lower bound for the area of Plateau foams

• Main Authors: Vicent Gimeno, Steen Markvorsen, José M. Sotoca
• Format: Article
• Language: English
• Published: SpringerOpen 2020-04-01
• Series: Journal of Inequalities and Applications
• Subjects: Foams; Bubbles; Density; Pressure; Comparison geometry
• Online Access: http://link.springer.com/article/10.1186/s13660-020-02362-4

Abstract Real foams can be viewed as geometrically well-organized dispersions 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.