FRG Collaborative Proposal: Eigenfunctions of the Laplacian

FRG 合作提案：拉普拉斯算子的本征函数

• 批准号: 0354539
• 负责人: Daniel Tataru
• 金额: $ 41万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354539&HistoricalAwards=false
• 关键词: FRG; Collaborative; Proposal; Eigenfunctions; Laplacian

Collaborative proposal DMS 0354386/0354539/0354668PIs: C.Sogge, S.Zelditch (Johns Hopkins)/D.Tataru, M.Zworski(U California, Berkeley)/H.Smith (U Washington)Title: Eigenfunctions of the LaplacianABSTRACTThis proposal is concerned with estimates of solutions of waveequations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are interested in how thegeometry, the boundary and the regularity of the metric influencecertain basic estimates. Problems of this kind arise both in thestudy of local and global existence of solutions of nonlinear waveequations and in the study of eigenfunctions of Laplacians inquantum chaos. Although these topics are widely separated in theirphysical origins, the relevant mathematics is closely related andthe techniques and insights in the two areas cross fertilize ina fruitful way. Each of the PIs is an expert one of these areas.By pooling our knowledge through a FRG grant we shall be able to make significant progress in these fields as wellas possibly in related ones such as general relativity theory.Eigenfunctions are the building blocks of functions. Understandingtheir basic size and concentration properties helps us solve new differential equations. In the other direction, techniques fromthe study of nonlinear wave equations have recently been usedto prove new results about eigenfunctions. This proposal isabout continuing this cross-fertilization of related fields.

合作提案DMS 0354386/0354539/0354668Pis：C.Sogge，S.Zelditch(Johns Hopkins)/D.Tataru，M.Zworski(加州大学伯克利分校)/H.Smith(U Washington)标题：拉普拉斯的本征函数摘要这个提案涉及(紧致和非紧致的)黎曼流形上波动方程的解的估计，可能是有边界的。我们感兴趣的是度量的几何、边界和正则性如何影响基本估计。这类问题既出现在研究非线性波动方程解的局部和整体存在性的问题中，也出现在研究量子混沌中的拉普拉斯本征函数中。虽然这些主题在物理起源上相去甚远，但相关的数学是密切相关的，这两个领域的技术和见解以卓有成效的方式交叉授粉。每一位PI都是这些领域的专家。通过FRG赠款汇集我们的知识，我们将能够在这些领域以及可能在相关领域如广义相对论取得重大进展。本征函数是函数的基础。了解它们的基本尺寸和浓度性质有助于我们求解新的微分方程式。另一方面，最近用研究非线性波动方程的方法证明了关于本征函数的新结果。这项建议是关于继续这种相关领域的交叉施肥。