Unique Continuation and Regularity of CR Mappings

CR映射的独特延续性和规律性

基本信息

批准号：
1855737
负责人：
Shiferaw Berhanu
金额：
$ 17.25万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855737&HistoricalAwards=false
关键词：
Unique Continuation Regularity CR Mappings

项目摘要

The main parts of the mathematics research project by Shiferaw Berhanu involve the investigation of the validity of unique continuation for solutions of systems of first order partial differential equations and second order partial differential equations. The project includes problems on the regularity of certain mappings between submanifolds in complex spaces. The second order partial differential equations under study arise in the study of electromagnetic radiation, optics, seismology, and acoustics. Some of the equations that are to be investigated are relevant to solid mechanics where they can be used to model elasto-static deformations. They are also of relevance in fluid mechanics since they can be employed to describe the motion of an incompressible viscous fluid. Results from the project have important applications to function theory of Several Complex Variables, CR Geometry, and linear as well as nonlinear partial differential equations. The project will provide several interesting problems to graduate students and young researchers.The first main problem concerns understanding geometric conditions on two Cauchy-Riemann submanifolds that guarantee unique continuation for a CR mapping between them that vanishes to infinite order at a point. The second problem concerns the unique continuation at the boundary for solutions of real analytic, second order or higher order, elliptic partial differential equations. The third problem involves the regularity of CR mappings between Cauchy-Riemann submanifolds. The methods to be employed include the theory of analytic discs, nonlinear Fourier transforms (FBI transforms), and a precise analysis of Green's functions for the second and higher order operators under study.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.

Shiferaw Berhanu的数学研究项目的主要部分涉及对一阶偏微分方程和二阶偏微分方程系统解决方案的独特延续有效性的研究。该项目包括有关复杂空间中子曼群之间某些映射的规律性的问题。所研究的二阶部分微分方程在电磁辐射，光学，地震学和声学的研究中出现。要研究的一些方程与可用于模拟弹性静态变形的固体力学有关。它们也与流体力学相关，因为它们可以用来描述不可压缩的粘性液的运动。该项目的结果具有重要的应用程序，可用于功能理论，该理论的几个复杂变量，CR几何形状和线性以及非线性偏微分方程。该项目将为研究生和年轻研究人员提供一些有趣的问题。第一个主要问题涉及了解两个Cauchy-Riemann Submanifolds上的几何条件，这些条件可以保证在某个时候消失到无限顺序的CR映射的独特延续。第二个问题涉及实现实际分析，二阶或更高阶，椭圆形偏微分方程的唯一延续。第三个问题涉及Cauchy-Riemann Submanifolds之间的CR映射的规律性。要采用的方法包括分析盘，非线性傅立叶变换（FBI变换）的理论，以及对正在研究的第二和高级运营商对格林功能的精确分析。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和宽广的影响来评估的支持，并被认为是值得的。