Problems in several complex variables and partial differential equations

主要研究者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，并在此过程中阐明n维复空间中与某个域相关的某些自然算子与该域边界对应算子之间的关系。本项目的一部分将解决d-bar Neumann问题的最大亚椭圆性问题，并分析具有亚椭圆边界拉普拉斯的（光滑，有界）伪凸域的Lp或Holder估计从内部转移到边界的问题。该项目还将关注当亚椭圆性不成立时的更退化的情况（即，将研究弱伪凸域上d-bar Neumann问题和边界拉普拉斯问题的正则性问题），特别是全局（ir）正则性，精确正则性和先验估计之间的联系。该项目将对以下广泛问题的答案做出重大贡献：给定域上的偏微分方程系统（具有规定的边界条件）的解的正则性与边界上相关系统的解的正则性是如何相关的？首席研究员介绍的一些方法应该可以应用于物理科学中出现的偏微分方程系统。在过去，对一些复杂变量的内切d-杆问题的研究常常导致分析方面的实质性进展，例如线性偏微分方程局部不可解的第一个例子的发现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。通过澄清复杂分析中出现的偏微分方程的一些不为人知的方面，这项研究可能会激发与数学和科学其他分支的新联系。

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.