Proposal: DMS-9870156 Principal Investigator: Mark S. Ashbaugh
Abstract: 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.
