# A lower Bound for the Area of Plateau Foams

## TL;DR

This paper establishes a new lower bound for the surface area of foam structures using a specialized divergence theorem, with applications to Kelvin foams and foam pressure differences.

## Contribution

It introduces a novel divergence theorem adapted to foam geometry and provides algorithms for estimating minimal foam surface areas based on point sets.

## Key findings

- Lower bounds for Kelvin foam cell surface area
- Lower bounds for foam cost function
- Estimation of pressure differences in minimal foams

## Abstract

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.