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

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.