Mathematical Sciences: Research on Hyperbolic Equations

摘要Sa Barreto 9623175这个项目分为四个主要部分：项目I拉普拉斯微扰的共振的存在性。项目二：史瓦西计量的共鸣。项目三.反双曲型问题.项目四.半线性波动方程解的奇性.项目I，PI将研究拉普拉斯二阶自伴微扰的共振的存在性。最近，调查员与M·兹沃斯基合作解决了这一问题的两个具体案例。我们还打算调查这两篇论文中使用的方法是否可以改进，以获得半径为r的圆盘中共振数的下界。项目II(与M.Zworski联合项目)PI将研究Regge-Wheeler方程的共振性。它模拟了没有质量和自转的引力场的扰动。我们的目的是研究是否可以改进Bcheot和Bcheot的工作，以证明Regge-Wheeler算子的预解式具有到整个复平面的亚纯扩张，并描述共振的位置。项目III.PI将考虑确定从Dirichlet到Neumann映射的势V的问题，对应于势V对拉普拉斯的扰动。我们提出了一种不涉及构造方程的指数递增解的方法，这是使用Calderon，Sylvester-Uhlmann等人的方法证明此类结果的主要困难。项目IV。(与Mark Joshi的联合项目)PI将构建半线性波动方程柯西问题的解的例子，初始数据垂直于光滑曲线，该曲线在燕尾点上的前光锥表面上是奇异的。我们打算使用固定相方法来实现这一点。这些项目的目的是确定微扰对介质的影响，并反过来，知道某种类型的微扰在介质中造成的影响，确定这种微扰的性质。这些问题都起源于物理学。在项目I中，PI想要确定介质中的扰动对声波通过该介质传播的影响。项目三涉及根据物体表面的测量结果确定该物体内部的材料的问题。PI将研究的问题是，如果两个物体有相同的测量，只在它们的表面上进行，那么它们一定是由相同的材料形成的。项目二是引力理论，人们想要更好地理解爱因斯坦场方程组的某些解。项目IV致力于某一介质的非线性微扰如何影响光在该介质中的传播的问题。

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.