Some new approaches for the study of properties of viscosity solutions

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This proposal concerns some nonlinear partial differential equations (PDE), which have deep connections with optimal control theory, game theory, mathematical finance, homogenization theory, and statistical physics. The main goal is to discover new underlying principles and generic methods to understand the properties of solutions of these nonlinear PDEs. One of the main objects of this proposed research is a class of non convex Hamilton--Jacobi equations, which are the fundamental equations for two-person, zero-sum differential games. Achieving deeper properties of their solutions (singular structures of the gradients, large time average, and so forth) will help a lot in the design of fast numerical methods to approximate the solutions accurately and in the design of optimal strategies for the players in the games.The proposed projects are to (i) continue developing a new approach to obtain large time behavior of solutions of Hamilton-Jacobi equations and related problems, (ii) discover game theory interpretation and dynamical properties of solutions of some weakly coupled systems, (iii) study homogenization of some Hamilton-Jacobi equations, and (iv) obtain a PDE approach to study asymptotic limit for the Langevin equation with vanishing friction coefficient. The topics consist of widely different nonlinear problems but they all satisfy maximum principle and hence admit viscosity solutions. The Crandall-Lions theory of viscosity solutions has been developed extensively in the last thirty years including the existence, uniqueness, stability of the solutions as well as some connections to differential games, front propagations, homogenization theory, optimal control, and weak KAM theory. However, many interesting properties of viscosity solutions, such as regularity, dynamical properties, gradient shock structure, and game theory interpretation of solutions, are still far from being well understood. The PI proposes to develop some new approaches to study (i)-(iv), which are expected to bring new perspective and insights to the field of viscosity solutions. The mathematical tools to be used for (i)-(iv) are composed by techniques from the nonlinear adjoint method (duality method), dynamical system, level set method, optimal control theory, and game theory.