Asymptotic Analysis of Geometric Partial Differential Equations

本项目涉及几何、物理和工程中出现的非线性椭圆偏微分方程。本课题研究的方程具有一定程度的简并性或奇异性。简并椭圆型方程是一类重要的非简并椭圆型方程，它与一致椭圆型方程有显著的区别。简并椭圆方程的解比一致椭圆方程的解具有更差的性质（通常是导数在一定程度上失去平滑性）。经典的一致椭圆方程分析，强调解的正则性（即导数的光滑性），对于退化椭圆方程的研究是不够的。该项目的主题是分析这些方程的解在退化或奇点发生的集合附近。主要任务之一是研究新格式解的性质，并确定解存在和正则性的最优条件。这样的任务反映在这个项目的所有问题中。本项目的核心部分是致力于研究退化奇异椭圆型微分方程解的性质的新方法和新技术的发展。除了研究几何和物理的基本问题外，该项目还有助于建立一个总体框架来研究几类重要的退化椭圆微分方程，并提高我们对这些方程的认识。学生培训和跨学科合作是这个项目的重要方面。3名数学研究生和1名电气工程研究生参与了这个项目。该项目将研究退化或奇异椭圆型偏微分方程。由于简并类型的复杂性和多样性，不可能对简并椭圆方程建立一个通用的理论。本项目主要研究在几何和物理中经常出现的三类简并椭圆方程。第一类是一致简并椭圆方程。许多重要的几何问题都归结为这类方程。这些方程定义在有边界的紧流形上，并且只在边界上简并，简并率与边界的距离一致。第二类方程由椭圆型方程组成，椭圆型方程在孤立点处奇异，其奇异率与到奇异点的距离一致。渐近展开式证明了这两类方程的解具有特定的奇异因子是奇异的。研究这些方程的主要目的是研究这些奇异因子对解的存在性和正则性的影响。第三类方程由椭圆方程和在余维数为2的集合中奇异的椭圆方程组组成。这类方程出现在爱因斯坦方程的径向对称解的研究中，接近极值克尔解。这类方程的研究将研究解在点附近的渐近行为，并证明质量不等式和角动量不等式。研究这三类方程需要完全不同的方法，尽管它们只是在发生退化的集合的维数上有所不同。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves nonlinear elliptic partial differential equations that arise in geometry, physics, and engineering. The equations studied in this project exhibit a certain degree of degeneracy or singularity. Degenerate elliptic equations demonstrate significant differences from uniformly elliptic equations, an important class of nondegenerate counterparts. Solutions of degenerate elliptic equations have much worse properties than those of uniformly elliptic equations (typically, a loss of smoothness of derivatives at some level). The classical analysis of uniformly elliptic equations, emphasizing the regularity of solutions (i.e., smoothness of their derivatives), is inadequate for studying degenerate elliptic equations. The theme of the project is the analysis of solutions to these equations near sets where degeneracy or singularity occurs. One of the main tasks is to study the properties of solutions in new formats and to identify optimal conditions for the existence and regularity of solutions. Such a task is reflected in all problems in this project. The central part of this project is dedicated to the development of new methods and techniques in studying the properties of solutions of degenerate and singular elliptic differential equations. In addition to pursuing fundamental problems in geometry and physics, this project also contributes to establishing a general framework to study several important classes of degenerate elliptic differential equations and advancing our knowledge of these equations. Student training and interdisciplinary collaborations are important aspects of this project. Three graduate students in mathematics and one graduate student in electrical engineering participate in this project. The project will study degenerate or singular elliptic partial differential equations. Due to the complexity and diversity of the types of degeneracy, it is impossible to develop a general theory for degenerate elliptic equations. The project focuses on three classes of degenerate elliptic equations that appear frequently in geometry and physics. The first class is the uniformly degenerate elliptic equations. Many important geometric problems are reduced to this class of equations. These equations are defined on compact manifolds with boundaries and degenerate only on boundaries, with a uniform rate of degeneracy in terms of the distance to the boundary. The second class of equations consists of elliptic equations that are singular at isolated points, with a uniform rate of singularity in terms of the distance to the singular point. Asymptotic expansions demonstrate that solutions to these two classes of equations are singular with specific singular factors. The primary objective of the research on these equations is to study the impact of these singular factors on the existence and regularity of solutions. The third class of equations consists of elliptic equations and systems of elliptic equations that are singular in a set of codimension 2. This class of equations appears in the study of radially symmetric solutions to the Einstein equation, near the extreme Kerr solution. The research on this class of equations will study the asymptotic behaviors of solutions near punctures and to prove the mass and angular momentum inequality. Completely different methods are required to study these three classes of equations, although they differ only in the dimension of the sets where degeneracy occurs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.