Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

在这个项目中所做的工作分为两类，由振荡积分理论引起的问题和由权重理论引起的问题。两者都与各种形式的偏微分方程分析有关。振荡积分反映了线性偏微分方程奇点的传播。利用fbi变换，可以确定复向量场的正则性，扩展了先前Hormander的结果。其他工作将集中于适用于波动方程边界问题的airy型积分的Lebesgue和Sobolev估计。对加权积分研究的进一步细化将用于沿Fefferman和Donnelly提出的路线获得对节点集和特征函数的估计。我们期望可以使用carleman型不等式来改进对特征函数的估计。本文还将对反边值问题进行进一步的研究，以寻找具有较大弱型范数的势的存在性反例。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义，并验证表达解决方案的方法。

Work done on this project divides into two categories, problems arising from oscillatory integral theory and those arising from the theory of weights. Both are tied to the analysis of partial differential equations in various guises. The oscillatory integrals reflect the propagation of singularities of linear partial differential equations. Using the FBI-transformation, work will be done in determining regularity of complex vector fields, extending earlier results of Hormander. Other work will focus on Lebesgue and Sobolev estimates for Airy-type integrals applicable to boundary problems for wave equations. Further refinement of research on weighted integrals will be used to obtain estimates on nodal sets and eigenfunctions along lines initiated by Fefferman and Donnelly. It is expected that one can use Carleman-type inequalities to improve estimates on the eigenfunctions. Additional work will done on inverse boundary value problems to seek an existence counterexample for the case where the potential has a large weak type norm. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.