Mathematical Sciences: Constrained Problems in the Calculus of Variations

数学科学：变分法中的约束问题

• 批准号: 8706782
• 负责人: David Kinderlehrer
• 金额: $ 4.64万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-15 至 1988-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8706782&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Constrained; Problems; Calculus

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.

建议提供为期三年的持续资助，以支持与连续物理问题相关的数学模型和方法。 所出现的微分方程组以及用于推导方程的相关变分原理通常是不稳定的、不完全定定的或不适定的。 将对这些问题的几个方面进行调查，需要采取多种方法。 从连续介质理论的角度来看，目标涉及有序材料中出现的稳定性、相变和缺陷。 适合这项工作的主要例子是液晶和结晶固体。 从数学上讲，人们希望获得某些连续统现象的分析解释，用约束和不变性群以及典型问题的解决方案的行为来表达。 特别令人感兴趣的是约束或群体的变化，导致它们成为相变的机制。 在这些情况下也可能表现出奇异的行为。 静态平衡配置的解对应于能量密度的驻点。 它们采用三维调和映射到球体的形式。 当对模型施加边界条件时，解中可能会出现奇点（文献中称为缺陷）。 缺陷在理解材料行为方面发挥着核心作用。 过去两年，许多研究人员开发了主要的新理论工作和初始数值程序。 目前的工作将继续分析缺陷理论并建立数值程序的验证。 其他工作将集中于解的唯一性和奇异集的大小（豪斯多夫维数）的测量。 这项研究是致力于晶体结构研究的更大努力的一部分，代表了复杂的数学思想和物理世界模型的令人印象深刻的结合。 首席研究员在澄清一些困难方面所采取的步骤令人印象深刻。 其他人现在也被吸引到这个地区。 毫无疑问，资助应该继续下去。