Some new approaches for the study of properties of viscosity solutions

本文讨论了一些非线性偏微分方程，这些方程与最优控制理论、博弈论、数学金融、均匀化理论和统计物理有着密切的联系。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解的性质。本研究的主要对象之一是一类非凸Hamilton—Jacobi方程，它是二人零和微分对策的基本方程。获得其解的更深层属性（梯度的奇异结构，大时间平均值等）将有助于设计快速数值方法来准确地近似解，并为游戏中的玩家设计最佳策略。拟开展的研究项目包括：(1)继续发展获得Hamilton-Jacobi方程及相关问题解的大时间行为的新方法；(2)发现一些弱耦合系统解的博弈论解释和动力学性质；(3)研究一些Hamilton-Jacobi方程的均匀化问题；(4)获得研究摩擦系数消失的Langevin方程渐近极限的PDE方法。这些题目包括各种各样的非线性问题，但它们都满足极大值原理，因此都有粘性解。近三十年来，粘性解的Crandall-Lions理论得到了广泛的发展，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，粘性解的许多有趣性质，如规律性、动力学性质、梯度激波结构和解的博弈论解释，仍然远远没有得到很好的理解。PI建议开发一些新的方法来研究(i)-(iv)，这些方法有望为粘度溶液领域带来新的视角和见解。用于(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.