Eigenvalue Problems in Mathematical Physics and Geometry

摘要：本研究项目的目的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。重点将放在膜的固定振动和自由振动的特征值问题以及夹紧板的振动和屈曲问题上。重点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)寻找低指标特征值比率的不等式，以及更一般地，用低特征值限定任意特征值的不等式；(3)从两个或多个问题中找到特征值的比较结果，或者将一个空间中的域的特征值与其他一些可能更好理解的空间中的域的特征值相关联。这些比较结果将包括那些在任意黎曼流形中表述的问题的给定特征值与更正则问题的相应特征值相关的结果，例如，恒定曲率空间（欧几里得空间，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，大部分的努力将集中在获得这些基本空间中特征值问题的锐界上，特别是在理解这些空间中测地线球的特征值问题上。研究上述问题的主要原因是，特征值与膜和板的振动特征频率（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使其屈曲的最小力）之间具有简单的关系。这些问题对物理学和工程学的重要性怎么强调都不为过。这里考虑了任意形状的膜和板，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，对于物理组件的设计来说，无论是将给定的固有频率调整到特定值，还是抑制组件在特定频率范围内的振动（例如，避免共振），都有很大的帮助。所考虑的特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意义。此外，应该有可能建立与适用于更一般微分算子的形式大致相同的特征值不等式；例如，薛定谔方程，它是量子物理学的基础。

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.