CAREER: Existence, regularity, uniqueness and stability in anisotropic geometric variational problems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Minimizing the area functional is one of the most famous examples of geometric variational problems in mathematics and it has had a major impact both in mathematics and in physics. However, for several natural phenomena, the area functional is only a first approximation. In order to capture microstructures, most models in the sciences use directionally dependent functionals, referred to as anisotropic energies. Ever since the introduction of anisotropic energies by Gibbs in the 19th century to model crystals, they have been extensively applied in material sciences and engineering, motivating seminal works in geometric analysis. However, since anisotropic energies are not invariant under translations and rotations, they don’t enjoy the conservation laws of the area functional, which makes them significantly more complicated to study. The PI will advance the anisotropic minimal surfaces theory. Understanding existence, regularity, uniqueness and stability of solutions to anisotropic geometric variational problems plays a major role in analysis, geometry, topology and physics. The PI will also conduct vertically integrated educational activities tied with the research activities. In particular, undergraduate and graduate students will be exposed to the problems and techniques of this project via the organization of seminars, conferences, and a summer school. The PI will study existence, regularity, uniqueness and stability properties of anisotropic minimal surfaces. The existence of anisotropic minimal surfaces in Riemannian manifolds will require extending the min-max theory. In order to determine the regularity of anisotropic minimal surfaces, the PI will study the related geometric nonlinear elliptic PDEs. In addition to the stationary configurations, this research will shed light on the anisotropic Brakke flow and its approximation, through the analysis of the related parabolic PDEs. This project will also address the uniqueness of critical points of the anisotropic isoperimetric problem and investigate the stability properties of the Wulff shapes. Furthermore, part of this research will be devoted to optimal transport, with an emphasis on the regularity and stability properties of branching dynamics. 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.