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: 8801218
• 负责人: David Tartakoff
• 金额: $ 4.45万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801218&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

Work planned on this project centers on a single broad question: isolate the local and global boundary regularity properties of partial differential operators on domains in complex n-dimensional space. The research combines analytic methods of differential equations with geometric concepts in several complex variables. The focus will be on differential operators which play a natural role in the study of holomorphic functions, the so-called d-bar and box-b operators. Aside from intrinsic interest, the research has its roots in a fundamental discovery of Charles Fefferman in the 1970's that biholomorphic mappings between certain domains (pseudoconvex) have natural and continuous extensions to the respective boundaries. To achieve this, an integral transform, the Bergman operator must be shown to be analytic on the boundary of each domain. This research seeks to expand the results to a much larger class of domains known as weakly pseudoconvex. One must begin with the case of domains with real analytic boundaries. The necessary estimates have been obtained to yield positive results in the case of certain weakly pseudoconvex domains, in particular domains embedded in larger, biholomorphic domains can be treated. New local results suggest that global analytic regularity up to the boundary is true and accessible. Work will concentrate on those domains which are nonembeddable, for if the conjecture fails, it must do so in one of these circumstances. Applications of this work are planned in the general classification of domains and in the development of a priori estimates of solutions of partial differential equations.

该项目计划的工作集中在一个广泛的 问题：分离局部和全局边界正则性 域上的偏微分算子的性质 n维复杂空间 该研究结合了分析 方法的微分方程与几何概念， 几个复杂的变量 重点将放在差异化 在研究全纯函数中起着自然作用的算子 函数，即所谓的d-bar和box-b运算符。 除了 内在的兴趣，研究有其根源，在一个基本的 查尔斯·费曼在20世纪70年代发现， 某些域之间的映射（伪凸）具有自然和 向各自的边界延伸。 实现 这是一个积分变换，Bergman算子必须被证明 在每个域的边界上解析。 这项研究旨在将结果扩大到更大的范围。 这类域称为弱伪凸域。 必须开始 具有真实的解析边界的域的情况。 的 已经获得了必要的估计，以产生积极的结果 在某些弱伪凸域的情况下，特别是 可以处理嵌入较大的双全纯域中的域。 新的局部结果表明，全球分析正则性高达 边界是真实的和可访问的。 工作将集中在 这些域是不可嵌入的，因为如果猜想 如果失败，它必须在这些情况之一。 这项工作的应用计划在一般 领域的分类和先验的发展 偏微分方程解的估计