Mathematical Sciences: Presidential Young Investigator Award

这个项目代表了一名总统青年研究员在偏微分方程式和几个复变量理论中出现的数学问题上所做的工作。在这项研究的背景下，这些不是单独的领域，而是通过与理解有关的基本问题联系在一起，该微分方程定义在一个区域的边界上，在一个由几个复变量组成的空间中，允许解。在这种情况下，边界数据作为全纯函数延伸到边界的邻域。刻画了两个复变量空间中可解问题的区域。目前的工作将着眼于更高维度的情况。有理由相信，直接概括到更高的维度可能是不可能的。在证明二维结果时所需的估计不直接结转。最近引入的新度量将提供对微分方程组基本解的良好估计，但无法获得所需的锐度。除了解决当今最紧迫的问题外，还将使用在分析流形上的偏微分算子方面开发的方法进行工作，流形不一定是几个复变量中区域的边界。

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.