Harmonic Analysis of Waves and Eigenfunctions

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This mathematics research project concerns the behavior of eigenfunctions and propagating waves in various settings where the methods of classical geometric optics do not apply. One focus of study is curved spaces where the Riemannian metric is of low regularity. For metrics which are not twice-differentiable the geodesic flow is not well-posed, and hence does not determine the flow of energy for waves on such spaces. We consider Lipschitz metrics (i.e. one bounded derivative), and show that one can nevertheless control the rate of dispersion of energy in a manner consistent with the uncertainty of the geodesic flow. We apply combinatorial arguments to obtain best possible bounds in Lebesgue spaces for eigenfunctions. Manifolds with boundary, where energy reflection off the boundary is specified by Dirichlet or Neumann conditions, are also studied. Here, diffraction effects and multiply reflected geodesics lead to violation of the standard rate of energy dispersion. We adapt tools developed for Lipschitz metrics, and combine them with reflection techniques to obtain optimal bounds, as well as new dispersive estimates for both wave and Schrodinger equations on manifolds with boundary. Application of these estimates include new well-posedness results for nonlinear wave and Schrodinger equations. A key method for all of our work is the decomposition of waves into appropriately scaled wave packets. We combine this decomposition with paradifferential techniques to obtain quantitative bounds on solutions in these settings, where the more precise asymptotic formulas of geometric optics do not hold.The study of vibrational modes and wave propagation finds applications in the equations of physics, in signal analysis, and in the seismic imaging methods used for geophysical exploration. This mathematics research project aims to establish quantitative bounds that are both of theoretical interest, for the behavior of high-frequency vibrations, and of practical interest, in particular for seismic imaging and for better understanding the nature of the errors that are introduced by computational approximations. One setting we study is that of rough media, characterized by the sound speed changing in a non-smooth manner, which is a model for the intricate mix of materials occurring within the earth. We also study the manner in which waves reflect off hard objects, including shapes which involve complex multiple reflections. In both of these settings, precise wavefront analysis methods do not exist. Instead, our work relies on decomposing signals into coherent wave-packets, and determining (to a suitable order of approximation) the evolution of these packets as they propagate through the media. The packets developed in our work have also found applications in medical imaging, including reducing the amount of scans needed to sense features of interest.