Mathematical Sciences: Partial Differential Equations in Complex Analysis

本课题主要研究偏微分方程学与多复变量理论之间的相互作用。这些互动的历史源远流长，成果丰硕，每个主题都向另一个主题提供了宝贵的信息。本文的工作建立在最近提出的多元非齐次Cauchy-Riemann方程的唯一延拓性质的基础上，该延拓性质将给出复空间中区域间全纯映射的边界行为的信息。将努力核实最重要类别的域名的属性。我们还将利用最近关于柯西-黎曼方程正则性的结果，结合几何论证来证明Bergman投影的正则性定理。这样的结果也适用于多个复变量的映射问题。用Szego投影和核表示平面区域上Laplace算子的Dirichlet和Neumann问题的经典解的程序刚刚完成。这些结果为数值计算偏微分方程组和保角映射中经典问题的解提供了新的实用方法。工作将继续探索Szego投影在相关问题上的应用。该项目的学生将被用来测试由此产生的数值方法的效率。多复变理论在线性偏微分方程组的分析中起着核心作用。它提供了一个很好的环境，在这个环境中，可以利用几何、分析和泛函分析的工具来处理与物理世界建模有关的方程。

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.