Mathematical Sciences: Research on Hyperbolic Equations

数学科学：双曲方程研究

• 批准号: 9623175
• 负责人: Antonio Sa Barreto
• 金额: $ 7.52万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623175&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Hyperbolic; Equations

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.

摘要 Sa Barreto 9623175 该项目分为四个主要部分： 项目 I. 拉普拉斯扰动共振的存在。项目二。史瓦西度量的共振。 项目三。 反双曲问题。项目 IV。半线性波动方程解的奇异性。 项目 I. PI 将研究拉普拉斯二阶自伴扰动的共振的存在性。研究人员最近与 M. Zworski 合作解决了这个问题的两个特殊案例。我们还打算研究是否可以改进这两篇论文中使用的方法以获得半径为 r 的圆盘中共振数量的下限。 项目二。 （与 M. Zworski 的联合项目）PI 将研究 Regge-Wheeler 方程的共振。它模拟了没有质量和自旋的引力场的扰动。 我们的目标是研究是否可以改进 Bachelot 和 Bachelot 的工作，以证明 Regge-Wheeler 算子的求解对整个复平面具有亚纯扩展，并描述共振的位置。项目三。 PI 将考虑从狄利克雷到诺伊曼映射确定势 V 的问题，该势对应于势 V 对拉普拉斯算子的扰动。我们提出了一种不涉及构造指数递增方程解的方法，这是使用 Calderon、Sylvester-Uhlmann 等人的方法证明此类结果的主要困难。项目四。 （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示例，初始数据与平滑曲线共轴，该曲线在燕尾点上方的前向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目标。 这些项目旨在确定扰动对介质的影响，并相互了解某种类型的扰动在介质中引起的影响，确定这种扰动的性质。这些问题起源于物理学。 在项目 I 中，PI 希望确定介质中的扰动对声波通过该介质传播的影响。 项目 III 涉及通过仅在给定物体表面进行的测量来确定该物体内部材料的问题。 PI 将研究的问题是，如果两个物体仅在其表面上进行了相同的测量，那么它们一定是由相同的材料形成的。 项目二是引力理论领域，希望更好地理解爱因斯坦场方程的某些解。项目 IV 致力于解决某种介质的非线性扰动如何影响光在该介质中的传播的问题。