Eigenvalue Problems in Mathematical Physics and Geometry

建议：DMS-9870156首席研究员：Mark S.Ashbaugh摘要：本研究项目的目标是建立数学物理和几何中出现的偏微分算子的特征值的界。重点讨论膜的固定振动和自由振动的特征值问题，以及固定板的振动和屈曲问题。集中将集中在以下具体问题上：(1)根据几何量找到这些问题的第一个或基本特征值的下界；(2)寻找低指数特征值的比率的不等式，以及更一般地，根据更低的特征值约束任意特征值的不等式；(3)寻找与两个或更多问题的特征值相关的比较结果，或者将一个空间中的区域的特征值与其他更好理解的空间中的区域的特征值相关联。在这些比较结果中将包括在任意黎曼流形中所表述的问题之一的给定特征值与更正则问题的相应特征值相关联的情况--例如，常曲率空间(欧几里得、球面或双曲空间)中球的相应特征值。由于它们所起的归一化作用，许多工作将致力于获得这些基本空间中本征值问题的锐界，特别是理解这些空间中测地球的本征值问题。研究上述问题的主要原因是，本征值与膜和板的振动特征(或固有)频率以及屈曲问题中的临界屈曲载荷(即，需要在板的周长周围施加的最小力)具有简单的关系。这些问题对物理学和工程学的重要性怎么强调都不为过。这里考虑任意形状的薄膜和板，并寻求根据几何参数控制特征频率和屈曲载荷的结果。特别是，人们通常最感兴趣的是膜或板的最低或所谓的基波频率，因为它通常是物理上最重要的频率。对各种几何特征如何影响这些频率有一个大致的了解，这对物理部件的设计有很大的帮助，无论是希望将给定的固有频率调谐到特定值还是希望抑制部件在特定频率范围内的振动(例如，为了避免共振)。所考虑的本征值问题也描述了波导的关键特性，因此对光纤具有重要的影响。此外，应该有可能建立适用于更一般的微分算符的基本相同形式的特征值不等式；例如，作为量子物理基础的薛定谔方程。

Proposal: DMS-9870156Principal Investigator: Mark S. AshbaughAbstract: The objective of this research project is to establish bounds on the eigenvalues of the partial differential operators which arise in mathematical physics and geometry. The emphasis will be on the eigenvalue problems for the fixed and free vibrations of a membrane and for the vibration and buckling of a clamped plate. Concentration will be focused on the following specific problems: (1) finding lower bounds for the first, or fundamental, eigenvalues of these problems in terms of geometric quantities; (2) finding inequalities for ratios of low-index eigenvalues and, more generally, inequalities bounding an arbitrary eigenvalue in terms of lower eigenvalues; (3) finding comparison results relating eigenvalues from two or more of the problems or relating eigenvalues for domains in one space to those in some other, perhaps better understood, space. Included in these comparison results would be those where a given eigenvalue of one of the problems formulated in an arbitrary Riemannian manifold is related back to the corresponding eigenvalue of a more regular problem -- for example, the corresponding eigenvalue of a ball in a constant curvature space (Euclidean, spherical, or hyperbolic space). Because of the normalizing role they play, much of the effort will be directed at obtaining sharp bounds for eigenvalue problems in these basic spaces, and particularly at understanding eigenvalue problems for geodesic balls in these spaces.The main reason for studying the problems described above is that eigenvalues bear a simple relation to the characteristic (or natural) frequencies of vibration of membranes and plates, as well as to the critical buckling load in the buckling problem (i.e., the minimum force that needs to be applied around the perimeter of a plate to make it buckle). The importance of such issues for physics and engineering cannot be overstated. Here membranes and plates of arbitrary shape are contemplated, and results that control the characteristic frequencies and buckling loads in terms of geometric parameters are sought. In particular, one is often most interested in the lowest, or so-called fundamental, frequency of a membrane or plate, since it is usually the most important one physically. Having a general understanding of how various geometrical features affect these frequencies can be a great aid in the design of physical components, whether the desire is to tune a given natural frequency to a specific value or to suppress vibrations of the component in a certain frequency range (to avoid resonance, for example). The eigenvalue problems considered also describe key properties of waveguides, and as such have important implications for fiber optics. Moreover, it ought to be possible to establish eigenvalue inequalities of much the same form that apply to more general differential operators; e.g., the Schroedinger equation, which is fundamental to quantum physics.