CAREER: Singular and Global Solutions to Nonlinear Elliptic Equations

University Of California-Irvine, Irvine CA

Investigators

Abstract

Physical laws and curvature conditions are written in the language of partial differential equations (PDE). Solutions to these equations are often singular (non-smooth), which limits the reliability of numerical approximations of solutions and presents significant challenges in their mathematical analysis. In this project, the PI will investigate the qualitative behavior of solutions to nonlinear elliptic PDE that play a central role in physics and geometry, with particular emphasis on the construction of singular and global examples. The project includes an educational component that involves researchers at many career stages through (a) the supervision of postdoctoral researchers and Ph.D. students; (b) the organization of a quarterly weekend conference to train students in scientific communication; (c) the organization of a winter workshop aimed at graduate and advanced undergraduate students, with week-long short courses by experts on research topics related to this project; and (d) the writing of a book based on advanced topics courses given by the PI, with significant input from the students who attended these courses. At a technical level, the main goals of the project are to (1) construct new examples of nonlinear entire solutions to variants of the minimal surface equation, and prove related Bernstein-type theorems; (2) investigate singular structures that appear in solutions to fully nonlinear elliptic equations such as the Monge-Ampere and quadratic Hessian equations, motivated by applications to complex geometry, optimal transport, and meteorology; and (3) construct new examples of singular minimizers of classical variational integrals in low dimensions, and discover structure conditions that prevent the formation of singularities. The equations under investigation share features (degenerate ellipticity and the existence of singular solutions) that limit the usefulness of standard techniques. To address these challenges, the PI will pursue new approaches that unite several areas of mathematics, and build sophisticated technical tools to carry out these approaches. 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.