Regularity and Singularity Issues in Geometric Variational Problems

Geometric variational problems are used to describe the behavior of systems driven by surface tension energies, like crystals or soap bubbles. Classical examples are the isoperimetric problem (find the solid of minimal perimeter enclosing a given volume) and the Plateau problem (finding the surface of minimal area spanning a given boundary curve). In the last 50-years, geometric variational problems have found a number of surprising applications in both pure and applied mathematics.

Their solutions can describe the equilibrium configurations of physical systems in Mathematical Physics, the behavior of black holes in General Relativity, or they can provide preferred representatives in homology and homotopy classes in Differential Topology. For solutions of geometric variational problems, the presence of singularities is unavoidable and it is linked to the physical behavior of the systems they model or to the concentration of topological obstructions in geometric problems.

A fine description of the regular and singular set is of fundamental importance in our understanding of the underlying problem. This project aims to enhance our understanding of the qualitative and quantitative behavior of solutions of geometric variational problems by addressing a series of basic questions concerning their regularity and the description of their singularities.

The answer to these questions will require the development of new methods and techniques, which will also be valuable in other areas of mathematics. The project will provide research training opportunities for graduate and undergraduate students.

Despite the great amount of study dedicated to geometric variational problems, several basic questions concerning the regular and singular behavior of their solutions are still poorly understood. This project is intended to address these questions and to develop new tools and techniques to tackle them. The project involves work on four deeply interconnected directions of research: boundary regularity for mass minimizing currents, regularity of critical points of anisotropic surface tensions, regularity of solutions to spectral optimization and free boundary problems, and the structure of PDE constrained measures.

Their study will require the interaction of techniques coming from different areas of mathematics, such as Partial Differential Equations (PDE), Geometric Analysis, Geometric Measure Theory, Topology, and Harmonic Analysis, as well as the introduction of new ones.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.