Problems in several complex variables and partial differential equations

主要研究人员将研究多维复分析中Cauchy-Riemann方程解的正则性的几个基本问题，并在此过程中澄清与n维复空间中的区域相关的某些自然算子与该区域边界上的对应算子之间的关系。这个项目的一部分将讨论d-bar Neumann问题的极大亚椭圆性，并分析具有次椭圆边界拉普拉斯的(光滑的、有界的)伪凸域的Lp或Holder估计从内部到边界的转换问题。该项目还将专注于次椭圆性不成立时更退化的情况(即，将研究d-bar Neumann问题的正则性问题和弱伪凸域上的边界拉普拉斯)，特别是全局(IR)正则性、精确正则性和先验估计之间的联系。这个项目将对以下广泛问题的回答做出重大贡献：给定区域上的偏微分方程组(具有规定的边界条件)的解的正则性与边界上的相关系统的解的正则性如何相关？首席研究员介绍的一些方法应该适用于物理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-bar问题的研究经常导致分析上的实质性进展，例如发现了线性偏微分方程局部不可解的第一个例子，以及伪微分算子的发展。此外，这样的问题与调和分析和代数几何有许多联系。通过澄清在复杂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出与数学和科学的其他分支的新联系。

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.