Mathematical Sciences: Constrained Problems in the Calculus of Variations

建议提供为期三年的持续资助，以支持与连续介质物理问题有关的数学模型和方法。出现的微分方程组和用于推导方程的相关变分原理通常是不稳定的、不完全适定的或不适定的。将对这些问题的几个方面进行调查，需要各种方法。从连续介质理论的观点来看，目标是关于有序材料的稳定性、相变和缺陷。适合这项工作的主要例子是液晶和晶体固体。在数学上，人们希望得到某些连续统现象的解析解释，用约束和不变群以及典型问题的解的行为来表示。特别值得关注的是约束或基团的变化，使它们成为相变的机制。在这些情况下，也可能表现出奇怪的行为。静态平衡位形的解对应于能量密度的驻点。它们以调和映射的形式进入三维球面。当边界条件施加到模型上时，解中可能出现奇点(在文献中称为缺陷)。缺陷在理解材料的行为方面起着核心作用。在过去的两年里，许多研究人员开展了重大的新理论工作和初步的数值计算程序。目前的工作将继续分析缺陷理论，并建立数值程序的有效性。其他工作将集中在解的唯一性和奇异集的大小(Hausdorff维度)的测量上。这项研究是致力于研究晶体结构的更大努力的一部分，代表了令人印象深刻的复杂数学思想和物理世界模型的结合。首席调查员在澄清一些困难方面采取的步骤令人印象深刻。其他人现在正被这一领域所吸引。毫无疑问，应该继续提供资金。

A three-year continuing grant is recommended in support of mathematical models and methods related to problems in continuum physics. The systems of differential equations which occur and the associated variational principles used to derive the equations are often unstable, incompletely posed or not well-posed. Investigations of several aspects of these problems, requiring an assortment of methods, will be carried out. Objectives, from the viewpoint of continuum theory, concern stability, phase transitions and defects which occur in ordered materials. Primary examples appropriate to this work are liquid crystals and crystalline solids. Mathematically, one wishes to obtain analytic interpretations of certain continuum phenomena, expressed in terms of constraints and invariance groups and the behavior of solutions for typical problems. Of special interest is the change of constraints or groups, leading them to become the mechanisms of phase transitions. Singular behavior may be exhibited as well in these circumstances. Solutions of the static equilibrium configuration correspond to stationary points of the energy density. They take the form of harmonic mappings into the sphere in three dimensions. When boundary conditions are placed on the model, singularities can arise in solutions (called defects in the literature). Defects play a central role in understanding the behavior of the material. Major new theoretical work and initial numerical procedures have been developed by a number of researchers in the past two years. The present work will continue analyzing the theory of defects and establishing validation of numerical procedures. Additional work will focus on uniqueness of solutions and measurements of the size (Hausdorff dimension) of the singular set. This research, part of a larger effort dedicated to the study of crystal structures, represents an impressive combination of mathematically sophisticated ideas and models of the physical world. The steps taken by the principal investigator in clarifying some of the difficulties has been impressive. Others are now being attracted to this area. Funding should be continued without question.