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.

部分微分方程描述了几个相互依赖数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讨与部分微分方程理论相关的两个显着问题。第一个主题是唯一延续的问题，它（粗略地）询问何时有关差分方程问题解决方案的无限数据以确定解决方案在宏观尺度上的行为。第二个主题是解决方案的规律性。在许多与物理相关的情况下，部分微分方程表现出自我改善的规律性：假定存在的解决方案在具有较弱定义的平稳性通知的广泛函数中存在，实际上可以显示后方以满足更加严格的平滑度条件。这样的结果对于部分微分方程的理论和科学应用都很重要。该项目中正在考虑的二阶偏微分方程的类型与其他物理意义的其他方程相关，例如波和热方程，klein-gordon方程，麦克斯尔方程和schrödinger方程。该项目还将通过研究生招聘和培训为科学识字劳动力的发展做出贡献。该项目的主要重点是CR歧管之间的CR映射理论。 CR映射是几个复杂变量理论中的基础概念。在这种情况下，一个普遍的独特延续问题询问何时在两个嵌入式CR歧管之间进行CR映射，该cr歧管在单个点上消失了无限顺序，必须在该点的附近消失。即使在简单的特殊情况下，也很少有足够的条件。这个问题将与具有实际分析系数的二阶均匀偏微分方程解决方案的解决方案进行研究。该项目的第二部分涉及嵌入式和抽象CR歧管上的Cr映射的规律性理论，以及在抽象CR歧管上的CR函数的规律性。这样的结果应用于一阶复杂非线性偏微分方程的系统解决方案的理论。当前的项目将集中于了解确保CR映射平稳性的基本歧管的条件。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为通过评估被认为是宝贵的支持。