Mathematical Sciences: Partial Differential Equations in Complex Analysis

本项目计划的工作集中在一个单一的广泛问题上：分离复杂n维空间域上偏微分算子的局部和全局边界正则性。该研究将微分方程的解析方法与若干复杂变量的几何概念相结合。重点将放在在全纯函数研究中扮演自然角色的微分算子，即所谓的d-bar和box-b算子。除了内在的兴趣之外，这项研究的根源在于查尔斯·费弗曼在20世纪70年代的一个基本发现，即某些域（伪凸）之间的生物全纯映射对各自的边界具有自然和连续的扩展。为了实现这一点，一个积分变换，Bergman算子必须在每个域的边界上被证明是解析的。这项研究试图将结果扩展到一个更大的领域，即弱伪凸。我们必须从具有实分析边界的域开始。对于某些弱伪凸域，特别是嵌入在较大的生物全纯域中的域，我们已经得到了必要的估计以得到肯定的结果。新的局部结果表明，直到边界的全局解析正则性是真实的和可接近的。工作将集中在那些不可嵌入的领域，因为如果猜想失败了，它必须在其中一种情况下失败。这项工作的应用计划在一般分类的领域和在发展的先验估计的解的偏微分方程。

Work planned on this project centers on a single broad question: isolate the local and global boundary regularity properties of partial differential operators on domains in complex n-dimensional space. The research combines analytic methods of differential equations with geometric concepts in several complex variables. The focus will be on differential operators which play a natural role in the study of holomorphic functions, the so-called d-bar and box-b operators. Aside from intrinsic interest, the research has its roots in a fundamental discovery of Charles Fefferman in the 1970's that biholomorphic mappings between certain domains (pseudoconvex) have natural and continuous extensions to the respective boundaries. To achieve this, an integral transform, the Bergman operator must be shown to be analytic on the boundary of each domain. This research seeks to expand the results to a much larger class of domains known as weakly pseudoconvex. One must begin with the case of domains with real analytic boundaries. The necessary estimates have been obtained to yield positive results in the case of certain weakly pseudoconvex domains, in particular domains embedded in larger, biholomorphic domains can be treated. New local results suggest that global analytic regularity up to the boundary is true and accessible. Work will concentrate on those domains which are nonembeddable, for if the conjecture fails, it must do so in one of these circumstances. Applications of this work are planned in the general classification of domains and in the development of a priori estimates of solutions of partial differential equations.