Partial Differential Equations in Several Complex Variables

肖美芝将研究偏微分方程的主题， 多复变函数论本文讨论了非光滑区域上的Cauchy-Riemann方程、切向Cauchy-Riemann方程以及它们与奇异积分和几何测度理论的相互关系。 Lipschitz域上的Cauchy-Riemann方程的研究包括复Neumann边值问题和Cauchy-Riemann方程的估计。函数理论将被分析 以及Bergman投影和双全纯映射。最近的结果调和分析Lipschitz域的主要工具。本文将继续研究切向Cauchy-Riemann方程整体解和局部解的正则性。 这包括在Lipschitz或高余维的CR流形上的存在性和正则性定理，也研究了抽象CR流形的同伦公式，Szego投影，Hodge理论和嵌入。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研究中发挥重要作用，这些问题是调和分析、几何测度理论、复几何和粗糙系数偏微分方程等现代分析中重要领域的交汇点。 调和分析是求解偏微分方程，特别是Dirichlet和Neumann边值问题的有力工具。 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。调和分析在现代已经发展成奇异积分理论和几何测度理论。 一个理论的发展影响着另一个理论，两者共同被视为数学中最重要和最优雅的理论之一。在多复变中，这些重叠的领域产生了更丰富和更深刻的结果，其影响塑造了现代偏微分方程，多复变和调和分析。他们的影响甚至超出了这些领域，进入其他领域，如复杂的微分几何，代数几何和数学物理。虽然在过去的几十年里，当域是光滑的情况下已经取得了很大的进展，当域是不太规则的知之甚少。 非光滑域上这些问题的研究已经是经典Dirichlet和Neumann边值问题调和分析研究的中心。建立在这些坚实的基础上，PI打算解决多个复杂变量中更具挑战性的问题。他们的解决方案将推进所有上述错综复杂的相关领域，并开辟了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.