Mathematical Sciences: Partial Differential Equations in Complex Analysis

这个项目计划的工作集中在一个广泛的问题上：隔离复n维空间中区域上偏微分算子的局部和全局边界正则性。该研究将微分方程组的解析方法与多复变量的几何概念相结合。重点将放在微分算子上，它们在全纯函数的研究中扮演着自然的角色，即所谓的d-bar和box-b算子。除了内在的兴趣之外，这项研究还源于查尔斯·费弗曼在1970年的《S》中的一个基本发现，即某些区域(伪凸)之间的双全纯映射具有到各自边界的自然且连续的延拓。为了实现这一点，必须证明Bergman算子在每个区域的边界上是解析的，这是一个积分变换。这项研究试图将结果扩展到一类更大的区域，即弱伪凸域。我们必须从具有真实解析边界的区域的情况开始。在某些弱伪凸域的情况下，特别是嵌入在较大的双全纯域中的区域可以被处理时，得到了必要的估计以产生正的结果。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达的。工作将集中在那些不可嵌入的域上，因为如果猜想失败，它必须在其中一种情况下这样做。这项工作计划在区域的一般分类和发展偏微分方程解的先验估计方面的应用。

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.