Eigenvalue Problems in Mathematical Physics and Geometry

提案：DMS-9870156主要研究者：Mark S. Ashbaugh摘要：本研究计画的目标是建立数学物理与几何中出现的偏微分算子的特征值的界。 重点将放在薄膜的固定和自由振动以及固支板的振动和屈曲的本征值问题上。集中将集中在以下具体问题：（1）寻找第一，或基本，这些问题的特征值的几何量的下界;（2）寻找不等式的比率低指标特征值，更一般地说，不等式界定一个任意特征值的下特征值;（3）找出两个或两个以上问题的特征值的比较结果，或者找出一个空间中的域的特征值与另一个可能更好理解的空间中的域的特征值的比较结果。 包括在这些比较结果将是那些在一个任意的黎曼流形制定的问题之一的给定的本征值是相关的回相应的本征值的一个更经常的问题-例如，相应的本征值的球在一个常曲率空间（欧几里德，球形，或双曲空间）。 由于它们所起的规范化作用，许多工作将致力于获得这些基本空间中特征值问题的精确界，特别是理解这些空间中测地线球的特征值问题。研究上述问题的主要原因是特征值与特征值之间有一个简单的关系。膜和板的振动（或自然）频率，以及屈曲问题中的临界屈曲载荷（即，需要围绕板的周边施加以使其弯曲的最小力）。这些问题对物理学和工程学的重要性怎么强调都不过分。 这里膜和板的任意形状的设想，并寻求控制的特征频率和屈曲载荷的几何参数方面的结果。特别是，人们通常对膜或板的最低频率或所谓的基频最感兴趣，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可以在物理组件的设计中有很大的帮助，无论是希望将给定的固有频率调谐到特定值还是抑制组件在特定频率范围内的振动（例如，以避免共振）。本征值问题也被认为是描述波导的关键属性，并因此对光纤光学有重要的影响。此外，应该有可能建立与适用于更一般的微分算子的形式大致相同的本征值不等式;例如，薛定谔方程，量子物理学的基础。

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.