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: 0354386
• 负责人: Christopher Sogge
• 金额: $ 40.7万
• 依托单位: Johns Hopkins University
• 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=0354386&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 (约翰霍普金斯大学)/D.Tataru, M.Zworski(加州大学伯克利分校)/H.Smith (华盛顿大学)标题：拉普拉斯算子的本征函数摘要该提案是涉及波动方程解的估计（两者紧致和非紧致）黎曼流形，可能有边界。 我们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。这类问题在非线性波动方程解的局部和全局存在性的研究以及量子混沌中拉普拉斯算子的本征函数的研究中都会出现。尽管这些主题在物理起源上存在很大差异，但相关的数学密切相关，并且这两个领域的技术和见解相互交叉，取得了丰硕的成果。 每个 PI 都是这些领域的专家。通过 FRG 拨款汇集我们的知识，我们将能够在这些领域以及可能在广义相对论等相关领域取得重大进展。本征函数是函数的构建块。 了解它们的基本尺寸和浓度特性有助于我们求解新的微分方程。 另一方面，非线性波动方程研究的技术最近被用来证明有关本征函数的新结果。 该提案旨在继续相关领域的交叉融合。