Partial Differential Equations and Several Complex Variables

摘要：Shaw计划继续她在复变函数理论中出现的偏微分方程的研究。特别地，我们将研究柯西-黎曼方程在Lipschitz域上的正则性以及在这些域上的泛函理论。这项工作代表了Lipschitz域上Dirichlet和Neumann边值问题的最新进展到非强制边值问题的自然扩展。为了获得解的最优正则性，将使用调和分析和先验估计。研究了Lipschitz域上全纯函数的边界行为、全纯函数的零和非光滑域上的Bergman投影等问题。此外，Shaw打算继续她对切向Cauchy-Riemann方程的研究，包括解的正则性和抽象CR结构的嵌入问题等主题。平方和算子和其他亚椭圆算子的边界正则性问题是本课题的第三个重点。受诸如热在固体介质中的传导等基本物理问题的启发，经典的狄利克雷和诺伊曼问题在近两个世纪以来一直是分析研究的中心。通过使用势理论、Schauder估计和奇异积分理论等技术工具，这些问题在光滑边界域的情况下已经得到了很好的理解。将这种理解扩展到边界不一定平滑的情况，这是在物理和工程问题中应用几个复杂变量时越来越频繁遇到的情况，是最近研究的一个活跃领域。事实上，采用几何测量理论、非光滑系数的Schauder估计和谐波测量的新方法已经被用来解决这些问题，这一过程导致了许多新领域和新方法的发展。在过去的三十年里，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，但是有所谓的“Lipschitz边界”。柯西-黎曼复合体的诺伊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是有趣的，而且在几个复杂变量的函数理论中也有很多应用——甚至更多。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.