Partial Differential Equations in Several Complex Variables

邵美琪将研究偏微分方程式的主题，这些偏微分方程式是由多个复变量的函数论产生的。本文讨论了三个具体领域的问题：非光滑区域上的柯西-黎曼方程、切向柯西-黎曼方程及其与奇异积分和几何测度论的相互作用。Lipschitz域上的Cauchy-Riemann方程包括复Neumann边值问题和Cauchy-Riemann方程的估计。我们将在强伪凸Lipschitz域上分析函数理论，以及Bergman投影和双全纯映射。关于Lipschitz域的调和分析的最新结果是主要的工具。切向柯西-黎曼方程整体解和局部解的正则性将继续研究。研究了高维CR流形上的同伦公式、Szego投影、Hodge理论以及抽象CR流形的嵌入问题。关于Lipschitz曲线和非光滑区域上的奇异积分理论和几何测度理论将在这些问题的解决中发挥重要作用。这些问题是调和分析、几何测度理论、复几何和具有粗糙系数的偏微分方程的交界点，都是现代分析中的重要领域。经典的调和分析为求解偏微分方程，特别是Dirichlet和Neumann边值问题提供了强有力的工具。调和分析也与一个复变量交织在一起，特别是对于单位圆盘中的函数论。在其现代版本中，调和分析已演变为奇异积分理论和几何测度论。其中一个的发展影响另一个，两者共同被认为是数学中最重要和最优雅的理论之一。在几个复变量中，这些重叠区域产生了更丰富、更深刻的结果，这些结果的影响形成了现代偏微分方程式、几个复变量和调和分析。他们的影响甚至超出了这些领域，延伸到其他领域，如复微分几何、代数几何和数学物理。虽然在过去的几十年里，对于域是光滑的情况已经取得了很大的进展，但当域不那么规则时，人们对此知之甚少。在非光滑区域上对这些问题的研究已经成为利用调和分析研究经典Dirichlet和Neumann边值问题的核心。在这些坚实的基础上，PI打算在几个复杂的变量中解决更具挑战性的问题。他们的解决方案将推进上述所有错综复杂的领域，并开辟一个广阔的未开发领域。

Mei-Chi Shaw will investigate topics in partial differential equationswhich arise from function theory in several complex variables. Problemsin three specific areas are discussed in this proposal : theCauchy-Riemann equations on nonsmooth domains, tangential Cauchy-Riemann equations and their interplay with singular integrals and geometric measure theory. Aspects of the Cauchy-Riemann equations on Lipschitz domains addressed include the complex Neumann boundary value problem and estimates of the Cauchy-Riemann equations. Function theory will be analyzed on strongly pseudoconvex Lipschitz domains, as well as the Bergman projectionand biholomorphic maps. Recent results in harmonic analysis on Lipschitzdomains are the main tools. Research on the regularity propertyof the global and local solutions of the tangential Cauchy-Riemannequation will be continued. This includes the existence and regularitytheorems on CR manifolds which are Lipschitz or of higher codimension.Homotopy formulas, Szego projection, Hodge theory and the embeddings ofabstract CR manifolds are also studied. Singular integral theory andgeometric measure theory on Lipschitz curves and nonsmooth domains willplay a major role in the approach to these problems.These problems are at the interface of harmonic analysis, geometric measuretheory, complex geometry and partial differential equations with rough coefficients, important fields all in modern analysis. Classically, harmonic analysis offers a powerful tool to solve partial differential equations, especially the Dirichlet and Neumann boundary value problems. Harmonic analysis is also intertwined with one complex variable, especially for function theory in the unit disc. In its modern version, harmonic analysis has evolved into singular integral theory and geometric measure theory. The development of one influences the other and both collectively are viewed as one of the most important and elegant theories in mathematics. In several complex variables, these overlapping areas have produced even richer and more profound results whose impact have shaped modern partial differential equations, several complex variables and harmonic analysis. Their influence even extends beyond these fields into other areas, like complex differential geometry, algebraic geometry and mathematical physics. While great progress has been made in the past few decades for the case when the domains are smooth, little is known when the domain is less regular. The investigation of these problems on nonsmooth domains is already central to the study of classical Dirichlet and Neumann boundary value problems using harmonic analysis. Built on these solid foundations, the PI intends to tackle more challenging problems in several complex variables. Their solution willadvance all the aforementioned intricately related fields and open up a vast unexplored area.