Partial Differential Equations in Several Complex Variables

邵美琪打算继续她对偏微分方程式的研究，偏微分方程式是由多个复变量的问题引起的。所讨论的问题包括：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理论的关系。描述了研究这些问题的新方法。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，并导致在复杂几何和几个复杂变量方面取得重大进展。P.I.将使用新技术，并与来自不同领域的数学家合作，调查这些问题。它将为每个领域带来新的视角和更深的深度。这些问题在应用数学和物理中也有应用。关于具有角和楔形区域的Hodge定理在电动力学和工程中有应用。它的起源可以追溯到经典的狄利克莱原理，也就是能量最小化问题的标准解。霍奇定理已经发展成为一个美丽的理论，并成为偏微分方程组、复分析、微分几何和调和分析中许多问题不可或缺的工具。建议的项目就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.