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

多复变量映射和函数的唯一连续性和正则性

• 批准号: 2323531
• 负责人: Shiferaw Berhanu
• 金额: $ 22.98万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-03-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2323531&HistoricalAwards=false
• 关键词: Unique; Continuation; Regularity; Mappings; Functions

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.

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、工程和其他科学领域的许多系统的数学建模中无处不在。该项目将在通过复数定义的高维空间的设置中探索与偏微分方程理论相关的两个突出问题。第一个主题是唯一连续性问题，粗略地说，它询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。第二个主题是解的规律性。在许多与物理相关的情况下，偏微分方程表现出自我改进的规律性特性：先验假设存在于具有弱定义平滑度概念的广泛函数中的解，实际上可以后验地证明满足更严格的平滑度条件。这些结果对于偏微分方程理论和科学应用都很重要。本项目中考虑的二阶偏微分方程类型与其他具有物理意义的方程相关，例如波方程和热方程、克莱因-戈登方程、麦克塞尔方程和薛定谔方程。该项目还将通过毕业生招聘和培训，为培养具有科学素养的劳动力做出贡献。该项目的主要重点是 CR 流形之间的 CR 映射理论。 CR 映射是多个复变量理论中的基本概念。在这种情况下，一个一般的唯一连续问题询问两个嵌入式 CR 流形之间的 CR 映射（在单个点处消失为无限阶）何时必须在该点的邻域中相同地消失。即使在简单的特殊情况下，已知的充分条件也很少。该问题将与具有实数解析系数的二阶次椭圆偏微分方程的解的相关唯一连续问题一起研究。该项目的第二部分涉及可嵌入和抽象 CR 流形上 CR 映射的正则理论，以及抽象 CR 流形上 CR 函数的正则性。这些结果可应用于一阶复杂非线性偏微分方程组的解理论。当前的项目将侧重于了解保证 CR 映射平滑的底层流形的条件。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。