PAIRED CALIBRATIONS APPLIED TO SOAP FILMS, IMMISCIBLE FLUIDS, AND SURFACES OR NETWORKS MINIMIZING OTHER NORMS
PAIRED CALIBRATIONS APPLIED TO SOAP FILMS, IMMISCIBLE FLUIDS, AND SURFACES OR NETWORKS MINIMIZING OTHER NORMS
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DOI：
10.2140/pjm.1994.166.55
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发表时间：
1994-11
影响因子：
0.6
通讯作者：
G. Lawlor;F. Morgan
中科院分区：
文献类型：
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作者：
G. Lawlor;F. Morgan
在本文中，我们介绍了一种新的方法来证明面积最小化，我们称之为“配对校准。“我们开始与最简单的应用，圆锥体的四面体，这出现在肥皂膜。然后，我们讨论不混溶的流体界面，晶体表面，和一维网络最小化其他规范。1.简介在她的分类肥皂膜奇点[T1]，让泰勒证明了只有通过消除的过程中，圆锥体的边缘正四面体之间的表面分离的四个面最小化的面积。我们给出了一个直接的证明，适用于在所有维度的正规单形。参见图1.0.1。几种不混溶流体的混合试图使与界面表面积成比例的能量最小化，但是比例常数对于每对流体是不同的。第二章证明了某些锥极小化这样的加权面积。晶体的表面能取决于方向，由单位法线上的范数Φ给出。第三章证明了某些锥Φ最小化，如三棱柱上的锥。假设涉及基本的几何问题，例如等边集合的可能基数的数量（即，两两等距点的集合）对于Rn上的范数。我们还考虑了可微范数Φ的1维Φ-最小化网络。众所周知，长度最小化网络以120°角分成三个。第四章对Rn中Φ-最小化网络的奇点进行了分类，并将n + 1确定为能在一点相交的线段数的精确界。
In this paper we introduce a new method for proving area-minimization which we call "paired calibrations." We begin with the simplest application, the cone over the tetrahedron, which appears in soap films. We then discuss immiscible fluid interfaces, crystal surfaces, and one-dimensional networks minimizing other norms. 1. Introduction In her classification of soap-film singularities [Tl], Jean Taylor proved only by the process of elimination that the cone over the edges of the regular tetrahedron minimizes area among surfaces separating the four faces. We give a direct proof which applies to regular simplices in all dimensions. See Figure 1.0.1. Configurations of several immiscible fluids try to minimize an energy proportional to interfacial surface area, but the constant of proportionality varies for each pair of fluids. Chapter 2 proves that certain cones minimize such weighted areas. The surface energy of a crystal depends on direction, as given by a norm Φ on unit normals. Chapter 3 proves certain cones Φminimizing, such as a cone over a triangular prism. The hypotheses involve basic geometric questions, such as the number of possible cardinalities of equilateral sets (i.e., sets of pairwise equidistant points) for a norm on Rn. We also consider 1-dimensional Φ-minimizing networks for differentiable norms Φ. It is well-known that length-minimizing networks meet in threes at 120° angles. Chapter 4 classifies the singularities in Φ-minimizing networks in Rn and establishes n + 1 as the sharp bound on the number of segments that can meet at a point.