Variable coefficient Fourier Analysis and its applications

变系数傅立叶分析及其应用

• 批准号: 0555162
• 负责人: Christopher Sogge
• 金额: --
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555162&HistoricalAwards=false
• 关键词: Variable; coefficient; Fourier; Analysis; its

ABSTRACTThis proposal is concerned with estimates of wave equations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are 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 Lebesgues-space estimates (both linear and bilinear) in space for eigenfunctions and quasi-modes, and (local or global) Strichartz estimates in space-time. 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 quasi-modes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics.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 I am especially interested in making further contributions to this area.

这一建议涉及(紧的和非紧的)黎曼流形上的波动方程的估计，可能是带边界的。我们感兴趣的是度量的几何、边界和正则性如何影响某些基本估计。这类问题出现在研究流形上的调和分析，研究非线性波动方程的局部解和整体解，以及研究量子混沌中的本征函数。虽然这些主题在物理和历史起源上相去甚远，但相关的数学是密切相关的。各个领域的技术和洞察力相互促进，成果丰硕。特别是，当前许多研究的一个共同主题(以及本提案中的问题)是试图理解和利用线性和非线性波动方程的本征函数和解的质量集中。我们考虑的基本估计是勒贝格空间估计(线性和双线性)在空间中的本征函数和准模，以及(局部或全局)时空中的Strichartz估计。主要问题围绕着几何，特别是边界的存在如何影响估计以及使估计饱和的解的种类。后一个问题与谱渐近中的模和准模的集中度、振动性和大小性质等问题密切相关(但仍不清楚)。在非紧致情况下，它还与共振的分布及其与受困大地坐标的关系密切相关。上述问题自然产生于数学与物理领域(包括广义相对论、量子力学和量子混沌)之间的相互作用。所采用的技术包括静止相和奇点传播的研究。在主要大学的量子物理小组中，有一组非常活跃的研究人员研究高能本征态，我特别感兴趣的是在这一领域做出进一步的贡献。