Some new approaches for the study of properties of viscosity solutions

这一建议涉及一些非线性偏微分方程（PDE），它与最优控制理论，博弈论，数学金融，均匀化理论和统计物理有着深刻的联系。主要目标是发现新的基本原理和通用方法来理解这些非线性偏微分方程的解的性质。本文的主要研究对象之一是一类非凸的汉密尔顿-雅可比方程，它是二人零和微分对策的基本方程。实现其解决方案的更深层次属性（梯度的奇异结构，大的时间平均，等）将有助于设计快速数值方法来精确地近似解，并为游戏中的参与者设计最优策略。建议的项目是（i）继续开发一种新的方法来获得汉密尔顿解的大时间行为-Jacobi方程及相关问题，（ii）发现了一些弱耦合系统解的博弈论解释和动力学性质，（iii）研究了一些Hamilton-Jacobi方程的均匀化，（iv）得到了研究摩擦系数为零的Langevin方程渐近极限的PDE方法。这些问题包括广泛不同的非线性问题，但他们都满足最大值原理，因此承认粘性解决方案。Crandall-Lions粘性解理论在过去的三十年中得到了广泛的发展，包括解的存在性、唯一性、稳定性以及与微分对策、前沿传播、均匀化理论、最优控制和弱KAM理论的一些联系。然而，粘性解的许多有趣的性质，如规则性，动力学性质，梯度激波结构，以及解的博弈论解释，仍然远未得到很好的理解。PI建议开发一些新的方法来研究（i）-（iv），这有望为粘度解决方案领域带来新的视角和见解。用于（i）-（iv）的数学工具由来自非线性伴随方法（对偶方法）、动力系统、水平集方法、最优控制理论和博弈论的技术组成。

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.