Regularity properties and singularity structure of solutions to linear and non-linear variational problems

线性和非线性变分问题解的正则性和奇点结构

• 批准号: 0707005
• 负责人: Neshan Wickramasekera
• 金额: $ 10.32万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707005&HistoricalAwards=false
• 关键词: Regularity; properties; singularity; structure; solutions

The main goal of the proposed project is to understand the nature of the branch point singularities of stable minimal hypersurfaces. The work proposed in this direction is aimed on the one hand at developing a local regularity theory for stable branched minimal hypersurfaces, generalizing the proposer's recent work in the case of multiplicity 2, and on the other hand at estimating the size and obtaining information concerning the structure of the set of their branch point singularities, focusing on multiplicity 2 case first. As a step in the regularity theory, and also as an interesting variational problem in its own right, it is also proposed to develop an independent theory for the corresponding "linear" variational problem; i.e. to study the regularity properties and the branching behavior of co-dimension 1 multiple valued critical points of the Dirichlet's integral.The research proposed here is directed towards reaching in two specific previously unexplored contexts a goal that is common to a broader class of problems; namely, understanding the critical points of functionals occurring in various mathematical and physical problems. One often faces the problem of finding an extremum of a mathematical or physical quantity such as volume or some energy. In order to find an extremum, one has to work in a sufficiently large space of competitors which often need to be allowed to carry certain undesirable properties (singularities), with the expectation that a critical point will have nicer behavior than a typical competitor in the class. It is indeed often the case that a critical point found this way has more regularity, but it is also typical that it carries some small set of essential singularities. In order to fully understand the nature of the critical point, it is therefore necessary to study its singularities as well as its behavior near the singularities.

该项目的主要目标是了解稳定最小超曲面的分支点奇点的性质。在这个方向上提出的工作一方面旨在发展稳定分支最小超曲面的局部正则理论，概括提议者最近在重数 2 的情况下的工作，另一方面估计其分支点奇点集的大小并获得有关其结构的信息，首先关注重数 2 的情况。作为正则理论的一个步骤，也作为其本身有趣的变分问题，还建议为相应的“线性”变分问题发展一个独立的理论；即研究狄利克雷积分的同维 1 多值临界点的正则性性质和分支行为。这里提出的研究旨在在两个特定的先前未探索的背景下实现更广泛的问题类所共有的目标；即理解各种数学和物理问题中泛函的临界点。人们经常面临寻找数学或物理量（例如体积或某种能量）的极值的问题。为了找到极值，必须在足够大的竞争者空间中工作，这些竞争者通常需要被允许携带某些不良特性（奇点），并期望临界点将具有比同类中典型竞争者更好的行为。确实，通常情况下，通过这种方式找到的临界点具有更多的规律性，但它也通常带有一些本质奇点。为了充分理解临界点的本质，有必要研究其奇点以及奇点附近的行为。