Nonlinear Partial Differential Equations

物理和工程科学中的基本方程大多是偏微分方程，其中最困难的是非线性方程。因此，数学的一个中心目标是发现广泛适用于许多重要的强非线性偏微分方程组的一般原理和方法。这一建议确定了几类有趣的此类问题：分析非凸Hamilton-Jacobi方程的解(及其在动态博弈论模型中的应用)，改进弱KAM理论的PDE方法(与经典动力学建立联系)，研究无穷拉普拉斯方程(极大范数变分问题的关键方程)解的正则性，以及研究某些强非线性抛物型偏微分方程组的解的正则性。这些不同的非线性方程具有很大不同的结构，但统一为变分、极大值原理和/或能量方法，尽管通常在某些奇异极限内。了解非线性偏微分方程解的存在、唯一性和正则性，并确定其行为，是基本的数学任务，鉴于这些方程的大量实际解释，具有有趣的意义。例如，非凸的Hamilton-Jacobi PDE是二人微分对策的基本方程，因此了解解的奇异结构直接有助于最优策略的设计。相关应用适用于所谓的平均场博弈理论，这是一种新发展的对大量竞争参与者的行为进行主题建模的理论。此外，设计快速而准确的数值方法依赖于对基本的非线性偏微分方程组的详细了解。大量的经验一再证明，在纯科学和应用科学中，具有数学自然结构的简单的非线性偏微分方程会出现和重现。

Most of the fundamental equations of the physical and engineering sciences are partial differential equations (PDE), of which the most difficult are nonlinear. A central goal of mathematics is therefore discovering general principles and methods widely applicable to the many important strongly nonlinear PDE. This proposal identifies several classes of interesting such problems: analyzing solutions of nonconvex Hamilton-Jacobi equations (with applications to dynamical game theory models), improving PDE methods for weak KAM theory (to establish connections with classical dynamics), studying the regularity of solutions of the infinity Laplacian equation (a key equation for max-norm variational problems), and investigating the regularity of solutions to certain strongly nonlinear parabolic systems of PDE. These various nonlinear equations have widely differing structure, but are unified as being accessible to variational, maximum principle and/or energy methods, although usually in certain singular limits.Understanding the existence, uniqueness and regularity of solutions to nonlinear partial differential equations (PDE), and ascertaining as well their behavior, are fundamental mathematical tasks, which have interesting implications in view of the numerous practical interpretations of these equations. For example, the nonconvex Hamilton--Jacobi PDE is the basic equation for two-person differential games; and so understanding the singular structure of solutions helps directly in the design of optimal strategies. Related applications apply to so-called mean field game theory, a newly developed subject modeling behavior of large populations of competing players. In addition the design of fast and accurate numerical methods depends upon detailed understanding of the underlying nonlinear PDE. Vast experience has repeatedly demonstrated that simple-looking nonlinear PDE, with mathematically natural structure, appear and reappear throughout the pure and applied sciences.