Problems in several complex variables and partial differential equations

多个复变量和偏微分方程的问题

• 批准号: 0654120
• 负责人: Kenneth Koenig
• 金额: $ 10.58万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654120&HistoricalAwards=false
• 关键词: Problems; several; complex; variables; partial

The principal investigator will study several basic questions concerning regularity properties of solutions to the Cauchy-Riemann equations in multidimensional complex analysis, and in the process he will clarify the relationship between certain natural operators associated with a domain in n-dimensional complex space and their counterparts on the boundary of that domain. One part of this project will address maximal hypoellipticity for the d-bar Neumann problem and analyze the problem of transferring Lp or Holder estimates from the interior to the boundary for (smooth, bounded) pseudoconvex domains with subelliptic boundary Laplacian. The project will also focus on the more degenerate situation when subellipticity does not hold (i.e., will investigate regularity issues for the d-bar Neumann problem and boundary Laplacian on weakly pseudoconvex domains), particularly the connections among global (ir)regularity, exact regularity, and a priori estimates. This project will make a significant contribution to the answer of the following broad question: How are the regularity properties of solutions to a system of partial differential equations (with prescribed boundary conditions) on a given domain related to the ones for an associated system on the boundary? Some of the methods introduced by the principal investigator should have applications to systems of partial differential equations that arise in the physical sciences. The study of the interior and tangential d-bar problems in several complex variables has in the past often led to substantial advances in analysis, such as the discovery of the first examples of local nonsolvability of linear partial differential equations and the development of pseudodifferential operators. Moreover, such problems have many connections to harmonic analysis and algebraic geometry. By clarifying some poorly understood aspects of partial differential equations that arise in complex analysis, this research may inspire new ties to other branches of mathematics and science.

首席研究者将研究有关多维复杂分析中解决方案解决方案的规律性特性的几个基本问​​题，在此过程中，他将阐明与n维复杂空间中与域与该域边界相关的某些自然操作员之间的关系。该项目的一部分将解决D-BAR Neumann问题的最大性低纤维化，并分析LP或持有人估计的问题，从内部到边界，用于（平滑，有限的）伪convex结构域，具有下层状边界边界laplacian。该项目还将集中于较堕落的情况时（即，将调查D-Bar Neumann问题的规律性问题，而边界Laplacian在弱伪convex域上，尤其是全球（IR）规律性，确切规律性和先验估计的联系。该项目将为以下广泛问题的答案做出重大贡献：在与边界上关联系统相关的给定域中，解决方程组（带有规定边界条件）的解决方案的规律性属性如何？主要研究者引入的一些方法应在物理科学中出现的部分微分方程系统应用。过去几个复杂变量中内部和切向D-bar问题的研究通常会导致分析的重大进展，例如发现线性部分微分方程的局部不辨别性的第一个例子以及伪差操作员的发展。此外，这些问题与谐波分析和代数几何形状有许多联系。通过澄清一些复杂分析中出现的部分微分方程的一些知识，这项研究可能会激发与数学和科学其他分支的新联系。