FRG Collaborative Proposal: Eigenfunctions of the Laplacian

合作提案DMS 0354386/0354539/0354668 PI：C.Sogge，S.Zelditch（Johns霍普金斯）/D.Tataru，M.Zworski（U加州，伯克利）/H.Smith（U华盛顿）题目：拉普拉斯算子的本征函数摘要这个提案涉及（紧致和非紧致）黎曼流形上波动方程解的估计，可能有边界。 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。这类问题出现在非线性波动方程解的局部和整体存在性研究以及Laplacian量子混沌的本征函数研究中。虽然这两个主题在物理起源上有很大的不同，但相关的数学是密切相关的，而且这两个领域的技术和见解以富有成效的方式交叉施肥。 每一个PI都是这些领域中的专家。通过FRG的资助，我们将能够在这些领域以及相关领域（如广义相对论）取得重大进展。本征函数是函数的基石。 了解它们的基本尺寸和浓度特性有助于我们解决新的微分方程。 在另一个方向上，非线性波动方程的研究方法最近被用来证明关于本征函数的新结果。 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.