Mathematical Sciences: Crystalline Variational Problems

Jean Taylor将继续她在最小能量表面上的研究。她将把注意力集中在最小化不具有方向均匀性的能量泛函上，这被称为各向异性。所使用的技术来自于变分学的一个分支，即几何测量理论，它对几何分析的许多领域产生了越来越大的影响。这项工作已应用于材料研究，并将继续与冶金学家约翰·卡恩（John Cahn）合作开展一些项目。她的主要目标之一是了解晶体型各向异性表面能最小化的几何结果。在一个稍微不同的方向上，她将尝试开发和使用面积最小化的组合技术。后一种技术适用于经常出现的多面体情况。能量泛函具有积分表示，其中被积量，即自由能，在单位球上定义。自由能反过来又产生了伍尔夫形状，这是泰勒大部分研究的对象。她将研究这些表面上几种新的奇点类型，以及在非均相成核中一些新的取向规则，这些规则在她早期与卡恩的一些工作中已经被预测到。其中一些将涉及允许积分中的边缘能量最小化。其他的研究将研究当能量函数允许变化时最小曲面的演化。

Jean Taylor will continue her work on surfaces of minimum energy. She will focus her attention on minimizing energy functionals which do not possess directional uniformity, this is known as anisotropism. The techniques to be used come from a branch of the calculus of variations, geometric measure theory, which is having an ever increasing impact on many areas of geometric analysis. This work has applications in materials research and some joint projects with John Cahn, a metallurgist, will be continued. One of her main objectives will be to understand the geometric consequences of minimization of crystal- type anisotropic surface energies. In a slightly different direction she will try to develop and employ combinatorial techniques of area minimization. These latter techniques become appropriate in the polyhedral situations which frequently arise. The energy functionals have an integral representation inwhich the integrand, known as the free energy, is defined on the unit sphere. The free energy in turn gives rise to the Wulff shape, which is the obect of most of Taylor's studies. She will study several new types of singularities in these surfaces and some new rules for orientations in heterogeneous nucleation, which have been predicted in some of her earlier work with Cahn. Some of this will involve allowing edge energies in the integral being minimized. Other investigations will study the evolution of the minimizing surfaces as the energy function is allowed to vary.