Mathematical Sciences: Partial Differential Equations in Complex Analysis

这个项目的重点是研究偏微分方程和几个复杂变量理论之间的相互作用。这些相互作用的历史是漫长而富有成效的，每个学科都为另一个学科提供了有价值的信息。目前的工作是建立在最近的非齐次柯西-黎曼方程的几个变量的唯一延拓性质的表述，这将产生关于复空间中域间全纯映射的边界行为的信息。将努力核实最重要的领域类别中的属性。我们还将利用柯西-黎曼方程的正则性的最新结果，结合几何参数来证明伯格曼投影的正则性定理。这样的结果也适用于几个复杂变量的映射问题。用Szego投影和核表示平面上域上拉普拉斯算子的Dirichlet和Neumann问题的经典解的程序已经完成。这些结果为偏微分方程和保角映射经典问题的数值计算提供了新的实用方法，这些方法在应用科学中很有意义。将继续探索Szego投影在相关问题上的应用。参与该项目的学生将被用来测试所得到的数值方法的效率。复变理论在线性偏微分方程的分析中起着重要的作用。它提供了一个很好的环境，在这个环境中，几何、分析和函数分析的工具可以用于与物理世界建模相关的方程。

This project focuses on the interaction between studies of partial differential equations and the theory of several complex variables. The history of these interactions has been long and fruitful, with each subject providing valuable information to the other. The present work builds on a recent formulation of a unique continuation property for the inhomogeneous Cauchy-Riemann equations in several variables that will yield information about boundary behavior of holomorphic mappings between domains in complex space. Efforts will be made to verify the property in the most important categories of domains. Work will also be done using recent results about the regularity of the Cauchy-Riemann equations combined with geometric arguments to prove regularity theorems for the Bergman projection. Such results also have applications to mappings problems in several complex variables. A program to express the classical solutions to the Dirichlet and Neumann problems for the laplace operator on domains in the plane in terms of the Szego projection and kernel has just been completed. These results give rise to new and practical methods for numerically computing solutions to classical problems in partial differential equations and conformal mapping which are of interest in the applied sciences. Work will continue exploring applications of the Szego projection to related problems. Students on the project will be used to test the efficiency of the resulting numerical methods. The theory of several complex variables plays a central role in the analysis of linear partial differential equations. It provides an excellent environment in which tools from geometry, analysis and functional analysis can be brought to bear on equations which relate to modeling of the physical world.