Nonlinear Partial Differential Equations

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project identifies for intensive study several classes of interesting problems concerning the properties of solutions of certain nonlinear partial differential equations (PDE). The specific topics are analyzing solutions of nonconvex Hamilton-Jacobi equations, improving PDE methods for weak KAM theory, studying regularity of solutions of the infinity Laplacian equation, developing PDE techniques for "inconsistent" stochastic optimal control problems, formulating variational principles for optimal symplectic maps, and studying certain strongly nonlinear parabolic systems. The various topics cited above have widely differing structures but are unified by reason of being accessible to either variational, maximum principle and/or energy methods, although usually in certain singular limits. Vast research experience has shown that simple-looking nonlinear partial differential equations, with mathematically natural structures, appear and reappear in the pure and applied sciences. In particular, most of the fundamental equations of the physical and engineering sciences are partial differential equations, of which the most difficult, and arguably most important, are nonlinear. So-called perturbation techniques can handle various "close-to-linear" equations, but there remains the real necessity of discovering general principles and methods for various important nonlinear equations in the large. Understanding the existence, uniqueness, and regularity of solutions, and ascertaining as well their behavior, are fundamental mathematical tasks that should have practical implications in view of the various interpretations of these equations. The particular equations singled out for study in the project are all natural in their structures, and consequently can be expected to arise in various applications. For instance, the nonconvex Hamilton--Jacobi equation is the basic equation for two-person, zero-sum differential games, so understanding the singular structure of its solutions helps directly in the design of optimal strategies. The proposed extension of partial differential equations methods to the nonstandard stochastic optimal control problems should likewise be fundamental in framing optimality conditions.