Partial Differential Equations and Several Complex Variables

提案：DMS-9801091主要研究者：Mei-Chi Shaw 摘要：肖计划继续她的调查偏微分方程中出现的理论职能的几个复杂的变量。特别地，将研究Lipschitz域上的Cauchy-Riemann方程的正则性以及此类域上的函数论。这项工作是一个自然的扩展最近的进展，狄利克雷和诺依曼边界值问题的Lipschitz域的边界值问题，是没有强制性的。调和分析和先验估计将被用来获得最佳的正则性的解决方案。有关问题的边界行为的全纯函数的Lipschitz域，零点的全纯函数，和伯格曼投影的非光滑域也将研究。此外，肖打算继续她的研究切柯西-黎曼方程，包括这些主题的解决方案和嵌入问题的规则性抽象CR结构。平方和算子和其他次椭圆算子的边界正则性问题构成了该项目的第三个焦点。 受固体介质中热传导等基本物理问题的启发，经典的狄利克雷和诺依曼问题在近两个世纪以来一直是分析研究的核心。通过使用的技术工具，如潜在的理论，Schauder估计，奇异积分理论，这些问题现在已经成为很好地理解的情况下，一个域的光滑边界。将这种理解扩展到边界不一定光滑的情况，这种情况在物理和工程问题的多个复变量应用中越来越频繁地遇到，一直是最近研究的一个活跃领域。事实上，新的方法采用几何测度理论，Schauder估计与非光滑系数，调和测度已employed来解决这些问题，一个过程，导致了许多新的领域和方法的发展。在过去的30年中，关于非光滑区域的椭圆边值问题已经获得了非常精确的结果，但具有所谓的“Lipschitz边界”。“柯西-黎曼复合体的诺依曼问题是这一领域的下一个重大挑战。它不仅从偏微分方程的角度来看很有趣，而且在多复变量函数论中也有许多应用-甚至更广泛。然而，只有注入重要的新思想，这个问题才能得到解决。

Proposal: DMS-9801091 Principal Investigator: Mei-Chi Shaw Abstract: Shaw plans to continue her investigation of partial differential equations that arise in the theory of functions of several complex variables. In particular, the regularity of the Cauchy-Riemann equations on Lipschitz domains and function theory on such domains will be studied. This work represents a natural extension of recent progress on the Dirichlet and Neumann boundary value problems on Lipschitz domains to boundary value problems that are not coercive. Harmonic analysis and a priori estimates will be used in order to obtain the optimal regularity property of the solution. Related problems on the boundary behaviour of holomorphic functions on Lipschitz domains, zeros of holomorphic functions, and Bergman projection on non-smooth domains will also be studied. In addition, Shaw intends to continue her research on the tangential Cauchy-Riemann equations, including such topics as the regularity of the solutions and embedding problems for abstract CR structures. Boundary regularity problems for sum- of-squares operators and other subelliptic operators constitute a third focal point of the project. Motivated by such fundamental physical problems as the conduction of heat through a solid medium, the classical Dirichlet and Neumann problems have for almost two centuries been central to the study of analysis. Through the use of technical tools like potential theory, Schauder estimates, and singular integral theory, these problems have now become quite well understood in the case of a domain with smooth boundary. The extension of this understanding to the setting where the boundary is not necessarily smooth, a situation encountered more and more frequently in applications of several complex variables to problems in physics and engineering, has been an active area of recent research. In fact, novel approaches employing geometric measure theory, Schauder estimates with non-smooth coefficients, and harmonic measure have been em ployed to tackle these problems, a process that has led to the development of many new fields and methods. Within the last thirty years very precise results have been obtained concerning elliptic boundary value problems for domains that are not smooth, but have so-called "Lipschitz boundaries." The Neumann problem for the Cauchy-Riemann complex presents the next great challenge in this area. Not only will it be interesting from the point of view of partial differential equations, but it will also have many applications in the function theory of several complex variables - and beyond. The problem will be solved, however, only with the injection of significant new ideas.