FRG Collaborative Proposal: Eigenfunctions of the Laplacian

合作提案DMS 0354386/0354539/0354668 pi: C.Sogge， S.Zelditch (Johns Hopkins)/D。塔塔鲁，M.Zworski（加州大学伯克利分校）/H。本文研究了可能有边界的黎曼流形（紧致和非紧致）上的波方程解的估计。我们感兴趣的是几何、边界和度规的规律性如何影响某些基本估计。这类问题出现在非线性波方程解的局部存在性和全局存在性的研究中，也出现在拉普拉斯量子混沌的本征函数的研究中。尽管这些主题在物理起源上有很大的区别，但相关的数学是密切相关的，并且这两个领域的技术和见解以富有成效的方式交叉施肥。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取得重大进展，也可能在广义相对论等相关领域取得重大进展。特征函数是函数的组成部分。了解它们的基本大小和浓度特性有助于我们求解新的微分方程。在另一个方向上，研究非线性波动方程的技术最近被用来证明关于特征函数的新结果。这个建议是关于继续这种相关领域的交叉施肥。

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.