Nonlinear Partial Differential Equations

本项目确定了几类与某些非线性偏微分方程解的性质有关的有趣问题进行深入研究。具体内容包括：分析非凸Hamilton-Jacobi方程的解，改进弱KAM理论的偏微分方程组方法，研究无穷拉普拉斯方程解的正则性，发展“不一致”随机最优控制问题的偏微分方程组技术，建立最优辛映射的变分原理，以及研究某些强非线性抛物型系统。上面提到的各种主题有很大不同的结构，但由于可以使用变分、最大值原理和/或能量方法而统一，尽管通常是在某些单一限制下。大量的研究经验表明，具有数学自然结构的简单的非线性偏微分方程式在纯科学和应用科学中出现和重现。特别是，物理和工程科学的大多数基本方程都是偏微分方程，其中最困难的，也可以说是最重要的，是非线性的。所谓的微扰技术可以处理各种“接近线性”的方程，但真正有必要发现各种重要的非线性方程的一般原理和方法。理解解的存在、唯一性和规律性，并确定它们的行为，是基本的数学任务，鉴于对这些方程的不同解释，应该具有实际意义。在该项目中被挑选出来研究的特定方程在其结构上都是自然的，因此可以预期在各种应用中出现。例如，非凸的Hamilton-Jacobi方程是二人零和微分对策的基本方程，因此了解其解的奇异结构直接有助于最优策略的设计。所提出的将偏微分方程组方法推广到非标准随机最优控制问题，同样应该是构建最优性条件的基础。

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.