Most of the fundamental equations of the physical and engineering sciences are partial differential equations (PDE), of which the most difficult are nonlinear. A central goal of mathematics is therefore discovering general principles and methods widely applicable to the many important strongly nonlinear PDE. This proposal identifies several classes of interesting such problems: analyzing solutions of nonconvex Hamilton-Jacobi equations (with applications to dynamical game theory models), improving PDE methods for weak KAM theory (to establish connections with classical dynamics), studying the regularity of solutions of the infinity Laplacian equation (a key equation for max-norm variational problems), and investigating the regularity of solutions to certain strongly nonlinear parabolic systems of PDE. These various nonlinear equations have widely differing structure, but are unified as being accessible to variational, maximum principle and/or energy methods, although usually in certain singular limits.
Understanding the existence, uniqueness and regularity of solutions to nonlinear partial differential equations (PDE), and ascertaining as well their behavior, are fundamental mathematical tasks, which have interesting implications in view of the numerous practical interpretations of these equations. For example, the nonconvex Hamilton--Jacobi PDE is the basic equation for two-person differential games; and so understanding the singular structure of solutions helps directly in the design of optimal strategies. Related applications apply to so-called mean field game theory, a newly developed subject modeling behavior of large populations of competing players. In addition the design of fast and accurate numerical methods depends upon detailed understanding of the underlying nonlinear PDE. Vast experience has repeatedly demonstrated that simple-looking nonlinear PDE, with mathematically natural structure, appear and reappear throughout the pure and applied sciences.