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.

主要研究人员将研究几个基本问题的正则性解决方案的柯西-黎曼方程在多维复分析，并在此过程中，他将澄清某些自然运营商与域之间的关系在n维复空间和他们的对应边界上的域。这个项目的一部分将解决d杆Neumann问题的最大亚椭圆性，并分析将Lp或保持器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亚椭圆率不成立时更退化的情况（即，将研究弱伪凸域上的d杆Neumann问题和边界Laplacian的正则性问题，特别是全局（ir）正则性，精确正则性和先验估计之间的联系。这个项目将作出重大贡献的答案以下广泛的问题：如何正则性的解决方案的偏微分方程系统（规定的边界条件）在一个给定的域相关的边界上的一个相关系统？一些方法介绍的主要研究者应该有应用系统的偏微分方程，出现在物理科学。在过去，对多个复变量的内部和切向d-杆问题的研究常常导致分析的实质性进展，例如发现线性偏微分方程的局部不可解性的第一个例子和伪微分算子的发展。此外，这类问题与调和分析和代数几何有许多联系。通过澄清在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的其他分支的新联系。