Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

在这个项目上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。两者都以不同的形式与偏微分方程的分析联系在一起。振荡积分反映了线性偏微分方程奇性的传播。利用FBI变换，将在确定复矢量场的正则性方面做工作，推广了Hormander的早期结果。其他工作将集中在适用于波动方程边值问题的Ariy型积分的Lebesgue型和Soblev估计。进一步改进加权积分的研究将被用来获得关于节点集和特征函数的估计，这些估计是由Fefferman和Donnelly发起的。期望人们可以利用Carleman类型的不等式来改进对特征函数的估计。对于势具有大的弱型范数的情况，我们将在反边值问题上做更多的工作来寻找存在的反例。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，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描述这些关系，而是从这些关系中提取定性和定量的意义，并验证表达解的方法。

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.