Spectral Problems in Geometry and Partial Differential Equations

几何和偏微分方程中的谱问题

• 批准号: 0710477
• 负责人: Peter Perry
• 金额: $ 13.99万
• 依托单位: University of Kentucky Research Foundation
• 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: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0710477&HistoricalAwards=false
• 关键词: Spectral; Problems; Geometry; Partial; Differential

This project will focus on three areas of spectral theory for partial differential equations: (1) resonances in geometric scattering, (2) harmonic analysis on two-step nilpotent Lie groups, and (3) inverse spectral methods in the theory of nonlinear dispersive equations. In the first project, the principle investigator will continue his work on spectral and scattering theory for asymptotically hyperbolic manifolds and complex manifolds, studying properties of scattering resonances and their relationship to underlying geometric invariants. In the second project, he will work toward understanding the asymptotic behavior of heat kernels and singularities of the wave trace for two-step nilpotent Lie groups. Nonlinear dispersive equations with singular initial data may be viewed, via the inverse scattering method, as linearized flows for the scattering data of a very singular potential. We hope to extend the inverse scattering picture for the KdV and mKdV equations to such singular data and obtain greater insight into these dynamical systems--by constructing the flow on Hilbert spaces of initial data which are singular but very natural from a dynamical point of view.Partial differential equations provide underlying mathematical models for such diverse physical phenomena as wave propagation and heat flow. An equally important part of any physical model is the geometry of the system it describes: for example, differently-shaped musical instruments produce sound waves with different frequencies, even though in all cases the production and propagation of waves is governed by the same differential equation. The first two projects above study the interaction between solutions of a certain partial differential equation and the geometry of the domain where the solutions are defined, and seek to relate quantifiable properties of the solutions to quantifiable properties of the underlying geometry. These projects are part of a larger effort in the mathematical community to understand "inverse problems" in which the geometry of a physical system is reconstructed from measurable data which are the solutions of a partial differential equation: seismology and medical imaging are among the areas of applied mathematics where such inverse problems occur. The third project is a contribution to the study of a class of equations arising in physics which describe the propagation of nonlinear waves by extending powerful solution methods to a richer set of data.

该项目将集中于部分微分方程的光谱理论领域的三个领域：（1）几何散射中的共振，（2）对两步nilpotent Lie组的谐波分析，以及（3）非线性分散方程理论的逆频谱方法。在第一个项目中，原理研究者将继续在渐近夸张的歧管和复杂的歧管上继续他在光谱和散射理论上的工作，研究散射共振的特性及其与潜在的几何不变性的关系。在第二个项目中，他将努力理解两步的尼尔氏谎言基团的热核的渐近行为和波形痕迹的奇异性。可以通过反向散射方法查看具有单数初始数据的非线性色散方程，作为非常奇异电位的散射数据的线性化流。我们希望将KDV和MKDV方程的反向散射图片扩展到这种奇异数据，并通过在初始数据的希尔伯特（Hilbert）空间上构建流程，从而使这些动态数据的流动构建流动，这些数据从动力学的角度来看，这些数据是非常自然的。从一个动力学的​​角度来看，这些方程为这种多样的物理现象提供了诸如波浪范围和热量的多样性物理现象的基本数学模型。 任何物理模型的同样重要的部分是它描述的系统的几何形状：例如，不同形状的乐器产生具有不同频率的声波，即使在所有情况下，波的产生和传播都由相同的微分方程控制。上面的前两个项目研究了定义解决方案的某个部分微分方程的解与域的几何形状之间的相互作用，并试图将溶液的可量化特性与基础几何形状的可量化特性联系起来。这些项目是数学社区中更大努力的一部分，以了解“反问题”，其中从可测量的数据中重建了物理系统的几何形状，这是部分微分方程的解决方案：地震学和医学成像是在发生这种反问题的应用数学领域之一。第三个项目是对物理学中产生的一类方程的贡献，这些方程通过将强大的解决方案方法扩展到一组更丰富的数据来描述非线性波的传播。