Mathematical Sciences: Research on Hyperbolic Equations

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Abstract Sa Barreto 9623175 This project is divided in four main parts: Project I. Existence of Resonances For Perturbations of the Laplacian. Project II. Resonances of the Schwarzschild Metric. Project III. Inverse Hyperbolic Problems.Project IV. Singularities of Solutions to Semilinear Wave Equations. Project I. The PI will investigate the existence of resonances for second order self-adjoint perturbations of the Laplacian. Two particular cases of this problem have been recently solved by the investigator in collaboration with M. Zworski. We also intend to investigate if the methods used in these two papers can be refined to obtain lower bounds for the number of resonances in a disk of radius r. Project II. (Joint project with M. Zworski) The PI will study the resonances for the Regge-Wheeler equation. It models the perturbation of a gravitational field without mass and spins. Our goal is to study whether one can refine the work of Bachelot and Bachelot to show that the resolvent of the Regge-Wheeler operator has a meromorphic extension to the whole complex plane and to describe the location of the resonances. Project III. The PI will consider the problem of determining a potential, V, from the Dirichlet to Neumann map corresponding to the perturbation of the Laplacian by the potential V. We propose a method that does not involve the construction of exponentially increasing solutions to the equation, which is the main difficultyin proving such results using the methods of Calderon, Sylvester-Uhlmann and others. Project IV. (Joint project with Mark Joshi) The PI will construct examples of solutions to a Cauchy problem for semilinear wave equations, with initial data conormal to a smooth curve, that is singular on the surface of the forward light cone over the swallowtail point. We intend to use stationary phase methods to accomplish this. The projects are aimed at determining the effects a perturbation has on a medium and reciprocally, knowing the effects a certain ty pe of perturbation causes in a medium, determine the nature of this perturbation. These problems have their origin in Physics. In project I the PI wants to determine the effects a perturbation in the medium has on the propagation of sound waves through that medium. Project III is related to the question of determining the material inside a given object from measurements made only on the surface of that object. The question the PI will study is if two objects have the same measurements, made only on their surfaces, then they must be formed by the same material. Project II is in gravitation theory, one would like to better understand certain solutions of Einstein's field equations. Project IV is dedicated to the question of how a non-linear perturbation of a certain medium affects the propagation of light in that medium.