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

该项目的主要目标是了解稳定极小超曲面的分支点奇点的性质。在这个方向上提出的工作的目的是一方面发展一个局部正则性理论的稳定分支极小超曲面，推广的提议者最近的工作的情况下，多重2，另一方面在估计的大小和获得有关的结构的一套分支点奇点，重点是多重2的情况下第一。作为正则性理论中的一个步骤，也作为一个有趣的变分问题本身，它也被建议为相应的“线性”变分问题发展一个独立的理论;即研究共聚合物的规整性和支化行为，维数1多值临界点的Dirichlet积分。这里提出的研究是针对达到在两个特定的以前未探索上下文是一个目标，是共同的一个更广泛的一类问题;即，理解临界点的泛函发生在各种数学和物理问题。人们经常面临的问题是找到一个数学或物理量，如体积或一些能量的极值。为了找到一个极值，人们必须在一个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期望的属性（奇点），期望临界点将比类中的典型竞争者具有更好的行为。的确，用这种方法找到的临界点往往具有更强的规律性，但它也通常带有一小部分本质奇点。为了充分理解临界点的性质，因此有必要研究它的奇点以及奇点附近的行为。

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.