Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

该项目的工作分为两类， 振荡积分理论和那些 由重量理论产生。 两者都与 分析各种形式的偏微分方程。 振荡积分反映了 线性偏微分方程的奇异性 使用 联邦调查局的转变，工作将在确定 复向量场的正则性，推广了 霍曼德 其他工作将集中在勒贝格和索博列夫 适用于边值问题的Airy型积分的估计 关于波动方程 进一步完善关于加权 积分将用于获得节点集的估计， 本征函数沿着线发起的费曼和唐纳利。 期望可以使用Carleman型不等式来 改进特征函数的估计。 额外的工作将 在逆边值问题上做的， 反例的情况下，潜在的有一个大的弱 类型规范 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义，并验证方法， 表达解决方案。

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.