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.

首席研究员将研究有关多维复分析中柯西-黎曼方程解的正则性的几个基本问​​题，并在此过程中阐明与n维复空间中的域相关的某些自然算子及其对应项之间的关系在该域的边界上。该项目的一部分将解决 d-bar 诺伊曼问题的最大亚椭圆性，并分析将 Lp 或 Holder 估计从内部转移到具有亚椭圆边界拉普拉斯算子的（光滑、有界）伪凸域的边界的问题。该项目还将重点关注次椭圆性不成立时的更退化情况（即，将研究弱伪凸域上的 d-bar Neumann 问题和边界拉普拉斯算子的正则性问题），特别是全局（ir）正则性、精确正则性之间的联系，以及先验估计。该项目将为回答以下广泛问题做出重大贡献：给定域上偏微分方程组（具有规定边界条件）的解的正则性与域上相关系统的解的正则性有何关系？边界？首席研究员引入的一些方法应该适用于物理科学中出现的偏微分方程组。过去，对多个复变量的内部和切向 d 杆问题的研究常常会带来分析方面的实质性进展，例如线性偏微分方程局部不可解性的第一个例子的发现以及伪微分算子的发展。此外，此类问题与调和分析和代数几何有很多联系。通过澄清复分析中出现的偏微分方程的一些鲜为人知的方面，这项研究可能会激发与数学和科学其他分支的新联系。