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

该项目的主要目的是了解稳定极小超曲面的分支点奇点的性质。这方面的工作一方面是发展稳定的分枝极小超曲面的局部正则性理论，推广了作者最近在重数为2的情况下的工作，另一方面是为了估计其分支点奇点集的大小和获得有关其结构的信息，首先集中在重数为2的情况。作为正则性理论的一步，同时也是一个有趣的变分问题本身，也被提出为相应的线性变分问题发展一个独立的理论，即研究Dirichlet积分的余维1多值临界点的正则性和分支行为。本文的研究目的是在两个特定的以前未被探索的背景下达到一个更广泛的问题的共同的目标，即理解在各种数学和物理问题中出现的泛函的临界点。人们经常面临寻找一个数学或物理量的极值的问题，例如体积或某个能量。为了找到极值，一个人必须在一个足够大的竞争对手空间中工作，这些竞争对手经常需要被允许携带某些不受欢迎的性质(奇点)，并期望临界点的行为将比班上典型的竞争对手更好。以这种方式找到的临界点往往更具规律性，但它带有一些小的本质奇点也是典型的情况。为了充分了解临界点的性质，有必要研究临界点的奇异性以及临界点在奇点附近的行为。

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.