Partial Differential Equations in Several Complex Variables

肖美芝打算继续她的调查偏微分方程，这产生的问题，在几个复杂的变量。本文讨论的问题包括：非光滑区域上的Cauchy-Riemann方程和切向Cauchy-Riemann复形，这些方程的几何性质，以及它们与复流形上函数论的关系。研究这些问题的新方法进行了说明。 切向Cauchy-Riemann方程的估计与抽象结构的嵌入定理和有界全纯函数的构造密切相关，这两个问题是多复变量和复几何中最具挑战性的问题。 他们的解决方案 将推进所有这些错综复杂的相关领域，并导致复杂几何和几个复杂变量的重大进展。私家侦探将使用新技术并与 不同领域的数学家。它将 为每个领域带来新的视角和更大的深度。这些问题在应用数学和物理学中也有应用。关于角域和楔形域的霍奇定理在电动力学和工程学中有应用。它的起源可以追溯到经典的狄利克雷原理，能量最小化问题的正则解。霍奇定理已经发展成为一个美丽的理论， 成为一种不可或缺的工具 应用于偏微分方程、复分析、微分几何和调和分析中的许多问题。拟议的项目是在这个方向上推进我们的知识。

Mei-Chi Shaw intends to continue her investigation of partial Differential equations, which arise from problems in several complex variables. The problems discussed in this proposal include: the Cauchy-Riemann equations and the tangential Cauchy-Riemann complex on Non-smooth domains, the geometric aspects of these equations with curvature terms, and their relations with function theory on complex manifolds. New approaches to study these problems are described. Estimates for the tangential Cauchy-Riemann equations are closely related to the embedding theorems of abstract structures and the construction of bounded holomorphic functions, two of the most challenging problems in several complex variables and complex geometry.The intellectual merit of the proposed activity stems from the fact that these problems are at the forefront of several complex variables and partial differential equations. Their solution will advance all these intricately related fields and lead to significant progress in complex geometry and several complex variables. The P.I. will investigate the problems using new techniques and in collaboration with mathematicians from different areas. It will bring new perspective and greater depth to each field. These problems also have applications in applied mathematics and physics. The Hodge theorem on domains with corners and wedges has applications in electrokinetics and engineering. Its origin goes back to the classical Dirichlet Principle, the canonical solution to the energy minimizing problem. The Hodge theorem has developed into a beautiful theory and becomes an indispensable tool to many problems in partial differential equations, complex analysis, differential geometry and harmonic analysis. The proposed project is to advance our knowledge in this direction.