Harmonic Analysis of Waves and Eigenfunctions

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Harmonic Analysis of Waves and EigenfunctionsAbstract of Proposed ResearchHart F. SmithThis project will study solutions of hyperbolic equations in the setting of metrics of low regularity, and the behavior of eigenfunctions for such metrics. Our primary efforts will be to obtain Lp norm bounds on solutions and eigenfunctions, and quantifying the dependence of these bounds on the Holder smoothness of the media. An important issue is controlling the propagation of energy for solutions of rough hyperbolic equations. For rough media the geodesic flow is ill-posed, and geometric optics methods break down. Prior research has shown that some localization of energy is possible in rough settings. In this project we shall attempt to obtain optimal results on the possible degree of energy concentration and the rate of dispersion. The key tool to be used is a wave packet decomposition that provides approximate representations of the solution. Frequency dependent scaling arguments will be invoked to establish the short time scales on which waves in rough media exhibit classical dispersion. The investigator will also adapt the above methods to the study of waves and eigenfunctions on manifolds with boundary. This is done by reflecting the metric across the boundary to obtain a Lipschitz metric on an open set. The geometry of the resulting geodesic flow combines with the short scale bounds to establish optimal Lp bounds for eigenfunctions on manifolds with boundary.This research project will investigate stationary vibrational modes and travelling waves in rough media; a rough medium being one where the physics which governs the speed of waves changes abruptly from point to point. This will be done using a representation of travelling waves as a sum of coherent pulses, each of which moves in a very simple fashion. The investigator is able to quantify the degree to which vibrational modes and waves can concentrate, depending on the roughness of the underlying media. Such results are important for the study of nonlinear wave interactions and questions of the outside observability of vibrations and waves. They also provide information on the reflection of waves off obstacles. We will study how multiple reflections increase the concentration of travelling waves, and also show that waves reflecting off convex obstacles disperse at similar rates to waves travelling without reflection.