Mathematical Sciences: Presidential Young Investigator Award

该项目代表了一个总统执行的工作 青年研究员对数学问题的出现， 偏微分方程理论和几个复 变量 在研究的背景下，这些不是分开的 但与下列基本问题有关的基本问题 当一个微分方程，定义在 多复变量空间中域的边界， 承认一个解决方案。 当这种情况发生时， 边界数据延伸到边界的邻域， 全纯函数 两个复空间中的Domain 可以解决问题的变量 表征了 目前的工作将着眼于更高的 维度案例 我们有理由相信， 可能不可能推广到更高的维度。 证明二维结果所需的估计值 而不是直接传递。 最近推出的新指标将 提供了基本解的良好估计， 微分方程，然而，一个人所需要的敏锐度不能被 得到了 除了解决这个最紧迫的问题外， 日，工作也将在使用开发的方法， 分析流形上的偏微分算子， 不一定是几个复杂域的边界 变量

This project represents work performed by a Presidential Young Investigator on mathematical problems arising in the theories of partial differential equations and several complex variables. In the context of the research these are not separate areas but are connected by fundamental questions relating to understanding when a differential equation, defined on the boundary of a domain in a space of several complex variables, admits a solution. When this is the case, it follows that the boundary data extends to a neighborhood of the boundary as an holomorphic function. Domains in the space of two complex variables for which the problem can be solved have been characterized. The present work will look at the higher dimensional cases. There is reason to believe that a direct generalization to higher dimensions may not be possible. Estimates needed in the proof of the two-dimensional result do not carry over directly. New metrics recently introduced will provide good estimates of fundamental solutions of the differential equations, however the sharpness one needs cannot be obtained. In addition to attacking this most pressing problem of the day, work will also be done in using methods developed in analyzing the partial differential operator on manifolds which are not necessarily the boundary of a domain in several complex variables.