Some new approaches for the study of properties of viscosity solutions

这一建议涉及到一些与最优控制理论、博弈论、数学金融学、齐次化理论和统计物理有着深刻联系的非线性偏微分方程组。主要目的是发现新的基本原理和一般方法来理解这些非线性偏微分方程解的性质。本文研究的主要对象之一是一类非凸的Hamilton-Jacobi方程，它们是二人零和微分对策的基本方程。获得其解的更深层次的性质(梯度的奇异结构、大时间平均等)将有助于设计快速的数值方法以精确逼近解，并有助于为博弈中的参与者设计最优策略。所提出的项目是：(I)继续发展一种新的方法来获得Hamilton-Jacobi方程及其相关问题的解的大时间性态；(Ii)发现一些弱耦合系统解的博弈论解释和动力学性质；(Iii)研究一些Hamilton-Jacobi方程的齐次化，得到了研究摩擦系数为零的朗之万方程的渐近极限的偏微分方程组方法。这些主题由截然不同的非线性问题组成，但它们都满足最大值原理，因此都允许有粘性解。在过去的三十年里，Crandall-Lions粘性解理论得到了广泛的发展，包括解的存在性、唯一性和稳定性，以及与微分对策、前沿传播、齐化理论、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，如正则性、动力学性质、梯度激波结构以及解的博弈论解释，仍远未被很好地理解。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。(I)-(Iv)的数学工具由非线性伴随方法(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巧组成。

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.