Linear and nonlinear problems in CR manifolds

CR 流形中的线性和非线性问题

• 批准号: 0714696
• 负责人: Shiferaw Berhanu
• 金额: $ 11万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0714696&HistoricalAwards=false
• 关键词: Linear; nonlinear; problems; CR; manifolds

This project focuses on the study of systems of linear first-order partial differential equations and nonlinear problems that exploit the results and methods of analysis on CR-manifolds. The systems of equations under investigation arise from locally integrable structures, a good model for which is a CR-manifold. The tools that are of potential use include the Baouendi-Treves approximation scheme, ideas from microlocal analysis (e.g., the FBI transform), analytic discs, the geometry of Sussmann's orbits, and the techniques developed in the theory of holomorphic extendability of CR-functions on semirigid CR-manifolds. Potential application areas for the research in this project include partial differential equations and geometry. Some of the nonlinear partial differential equations under consideration surface in meteorology, in attempts to model and explain atmospheric phenomena. The same equations describe the evolution of the amplitudes of nonlinear waves in elastic solids. Understanding the propagation and interaction of nonlinear waves is a very important and challenging problem in physics. In geometry and physics, these equations arise in the study of the shapes of surfaces in space that turn up as limit shapes in a class of random surface models, limit shapes that minimize surface tension. The research activity is also expected to generate interesting problems for graduate students.

本项目主要研究一阶线性偏微分方程组和利用CR流形上的分析结果和方法的非线性问题。所研究的方程组源于局部可积结构，其中一个很好的模型是CR-流形。可能有用的工具包括Baouendi-Treves近似格式、微局部分析的思想(例如FBI变换)、解析圆盘、Sussmann轨道的几何，以及在半刚性CR-流形上CR-函数的全纯可扩性理论中发展起来的技术。该项目研究的潜在应用领域包括偏微分方程组和几何。气象学中考虑的一些非线性偏微分方程组，试图模拟和解释大气现象。同样的方程描述了弹性固体中非线性波的振幅的演变。了解非线性波的传播和相互作用是物理学中一个非常重要和具有挑战性的问题。在几何学和物理学中，这些方程是在研究空间中曲面的形状时产生的，这些曲面的形状在一类随机表面模型中表现为极限形状，即使表面张力最小化的极限形状。这项研究活动还有望为研究生带来有趣的问题。