Unique Continuation and Regularity of Mappings and Functions in Several Complex Variables

偏微分方程描述了几个相互依赖的量之间的动态相互关系。因此，它们在物理学、工程学和其他科学领域的大量系统的数学建模中无处不在。这个项目将探讨与偏微分方程理论相关的两个突出问题，在通过复数定义的高维空间中。第一个主题是唯一延拓的问题，粗略地说，就是问微分方程解的无穷小数据何时足以确定解在宏观尺度上的行为。第二个主题是解的正则性。在许多物理相关的情况下，偏微分方程表现出自我改进的正则性：先验假设存在于具有弱定义光滑性概念的广泛函数类中的解，实际上可以证明后验满足更严格的光滑性条件。这些结果对于偏微分方程理论和科学应用都是重要的。在这个项目中考虑的二阶偏微分方程的类型与其他具有物理意义的方程有关，例如波动方程和热方程，Klein-Gordon方程，Maxell方程和薛定谔方程。该项目还将通过招聘和培训研究生来促进科学素养劳动力的发展。该项目的主要重点是CR流形之间的CR映射理论。CR映射是多复变理论中的一个基本概念。在这种情况下，一个一般的唯一延续问题问，当两个嵌入的CR流形之间的CR映射，在一个单一的点消失到无限阶，必须消失在该点的邻域相同。很少有充分条件是已知的，即使在简单的特殊情况下。这个问题将与二阶次椭圆型偏微分方程解的相关唯一延拓问题一起研究，这些方程具有真实的解析系数。项目的第二部分涉及可嵌入和抽象CR流形上CR映射的正则性理论，沿着抽象CR流形上CR函数的正则性。这些结果对一阶复非线性偏微分方程组的解的理论有应用。目前的项目将侧重于了解基础流形上的条件，以保证CR映射的平滑性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Partial differential equations describe dynamic interrelationships between several mutually dependent quantities. As such, they are ubiquitous in the mathematical modelling of a host of systems in physics, engineering, and other scientific fields. This project will explore two salient issues related to the theory of partial differential equations, in the setting of higher-dimensional spaces defined via complex numbers. The first topic is the question of unique continuation, which – in rough terms – asks when infinitesimal data about the solution to a differential equation suffices to determine the behavior of the solution on macroscopic scale. The second topic is the regularity of solutions. In many physically relevant situations, partial differential equations exhibit a self-improving regularity property: solutions that are a priori assumed to exist within a broad class of functions with a weakly defined notion of smoothness, in fact can be shown a posteriori to satisfy a much more stringent smoothness condition. Such results are important both for the theory of partial differential equations and for scientific applications. The types of second order partial differential equations under consideration in this project are relevant for other equations of physical significance, such as the wave and heat equations, the Klein-Gordon equation, Maxell’s equation, and Schrödinger’s equation. The project will also contribute to the development of a scientifically literate workforce through graduate recruitment and training.The primary focus of the project is on the theory of CR mappings between CR manifolds. CR mappings are a foundational concept in the theory of several complex variables. In this context, a general unique continuation problem asks when a CR mapping between two embedded CR manifolds, that vanishes to infinite order at a single point, must vanish identically in a neighborhood of that point. Very few sufficient conditions are known, even in simple special cases. This question will be investigated in concert with related unique continuation problems for solutions of second order subelliptic partial differential equations with real analytic coefficients. The second part of the project involves the regularity theory of CR mappings on embeddable and abstract CR manifolds, along with the regularity of CR functions on abstract CR manifolds. Such results have applications to the theory of solutions of systems of first order complex nonlinear partial differential equations. The current project will focus on understanding conditions on the underlying manifolds that guarantee smoothness of CR mappings.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.