Mathematical Sciences: Partial Differential Equations in Complex Analysis

数学科学：复分析中的偏微分方程

• 批准号: 9302513
• 负责人: Steven Bell
• 金额: $ 17.4万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302513&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

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.

该项目的重点是研究之间的相互作用 偏微分方程与多复形理论 变量 这些相互作用的历史由来已久， 成果丰硕，每一个主题都提供了宝贵的信息， 其他.目前的工作建立在最近制定的一个 非齐次Cauchy-Riemann方程的唯一延拓性质 几个变量的方程，将产生关于 域间全纯映射的边界性态 复杂空间 将作出努力， 最重要的领域。 工作也将完成 利用最近关于柯西-黎曼正则性的结果， 结合几何论证证明正则性的方程 伯格曼投影的定理 这样的结果也有 应用于多复变量映射问题。一 程序来表达经典的解决方案狄利克雷和 平面区域上拉普拉斯算子的Neumann问题 在Szego投影和内核方面， 完成 这些结果产生了新的和实用的方法 用于数值计算经典问题的解决方案， 偏微分方程和保角映射， 对应用科学感兴趣。 工作将继续探索 Szego投影在相关问题中的应用 该项目的学生将被用来测试的效率， 产生的数值方法。 多复变量理论在其中起着核心作用 在线性偏微分方程的分析中。 它 提供了一个极好的环境， 分析和功能分析可以用来 与物理世界建模相关的方程。