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.

中文翻译：

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高原泡沫面积的下限

真正的泡沫可以看作是在液体中或多或少的球形气泡的几何结构良好的分散体。当泡沫排干从而液体含量显着减少时，泡沫就会变成多面体状，并且根据著名的高原定律，泡沫现在可以看作是在高原边界彼此相交的薄液膜网络。在本文中，我们从下面的球形球形泡沫的表面积下进行估算。我们的主要工具是发散定理的新版本，它适合于泡沫的特定几何形状，并特别注意其经典的高原奇点。作为我们结果的基准应用，我们获得了开尔文泡沫的基本孔的下界，所谓的成本函数的下界，并且针对最小周期泡沫中出现的压力差异。此外，我们提供了一种算法，其输入是空间中的一组孤立点，其输出是包含给定集合作为其顶点集的泡沫区域的最佳下界估计。