Variable Coefficient Fourier Analysis

变系数傅立叶分析

• 批准号: 1069175
• 负责人: Christopher Sogge
• 金额: $ 37.49万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1069175&HistoricalAwards=false
• 关键词: Variable; Coefficient; Fourier; Analysis

This research is concerned with estimates of wave equations on (both compact and noncompact) Riemannian manifolds, possibly with boundary. The investigator is interested in how the geometry, the boundary and the regularity of the metric influence certain basic estimates. Problems of this kind arise in the study of harmonic analysis on manifolds, the study of local and global solutions of nonlinear wave equations and in the study of eigenfunctions in quantum chaos. Although these topics are widely separated in their physical and historical origins, the relevant mathematics is closely related. Techniques and insights in the various areas cross-fertilize each other in a fruitful way. In particular, a common theme of much current research (and the problems in this proposal) is to try to understand and exploit the mass concentration of eigenfunctions and solutions of linear and nonlinear wave equations. The basic estimates that we have in mind are Lebesgue-space estimates (both linear and bilinear) in space for eigenfunctions and quasimodes, and (local or global) Strichartz estimates in spacetime. The main questions center around how the geometry and especially the presence of a boundary affects the estimates and the kinds of solutions that saturate them. The latter issue is closely related to the much studied (but still not well understood) questions of concentration, oscillation and size properties of modes and quasimodes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics. In the compact setting we are also interested in exploring how concentration properties of eigenfunctions are related to their nodal sets, which is the set of points where the function vanishes. The investigator wishes to use current and new estimates for eigenfunctions to make further progress on Yau's conjecture about the size of these sets. He also is interested in seeing how certain curvature assumptions affect these estimates and exploring connections with quantum ergodicity.The above problems arise naturally from interactions between mathematics and areas in physics that include general relativity, quantum mechanics, and quantum chaos. The techniques employed include stationary phase and the study of propagation of singularities. There is a very active group of researchers in quantum physics groups at major universities studying high-energy eigenstates, and the investigator is especially interested in making further contributions to this area.

这项研究涉及（紧致和非紧致）黎曼流形上的波动方程的估计，可能带有边界。 研究者感兴趣的是度量的几何形状、边界和规律性如何影响某些基本估计。 此类问题出现在流形调和分析的研究、非线性波动方程的局部和全局解的研究以及量子混沌中的本征函数的研究中。 尽管这些主题的物理和历史起源相去甚远，但相关的数学却密切相关。 各个领域的技术和见解相互交叉，取得了丰硕的成果。 特别是，当前许多研究（以及本提案中的问题）的一个共同主题是尝试理解和利用特征函数的质量集中以及线性和非线性波动方程的解。我们想到的基本估计是本征函数和准模态的勒贝格空间估计（线性和双线性），以及时空的（局部或全局）Strichartz 估计。 主要问题集中在几何形状，尤其是边界的存在如何影响估计以及使估计饱和的解决方案的种类。 后一个问题与谱渐进中模态和准模态的浓度、振荡和尺寸特性等经过大量研究（但尚未充分理解）的问题密切相关。 在非紧环境中，它也与共振的分布及其与俘获测地线的关系密切相关。 在紧凑的设置中，我们还有兴趣探索特征函数的浓度特性如何与其节点集相关，节点集是函数消失的点集。研究人员希望利用当前和新的特征函数估计，在丘关于这些集合大小的猜想上取得进一步进展。 他还对了解某些曲率假设如何影响这些估计以及探索与量子遍历性的联系感兴趣。上述问题自然产生于数学与物理学领域（包括广义相对论、量子力学和量子混沌）之间的相互作用。 所采用的技术包括固定相和奇点传播研究。 各大大学的量子物理组中有一群非常活跃的研究人员正在研究高能本征态，研究人员特别有兴趣在这一领域做出进一步的贡献。