Partial Differential Equations

建议：DMS-9801626首席研究员：约瑟夫·J·科恩摘要：科恩将继续研究几个复变量理论、偏微分方程及相关课题。特别是，他感兴趣的是：整体正则性，亚椭圆性，Holder和L-p估计，具有分段光滑边界的区域，以及CR流形的嵌入。科恩目前正在研究亚椭圆性问题，包括几个复变量的边界行为，以及微分和伪微分算子。他的兴趣是找到亚椭圆性成立的定性条件(无论是几何条件还是代数条件)。这项研究的技术部分与不“获得”平稳性的估计有关，因此误差项与试图估计的对象具有相同的幅度。科恩正在开发处理这类困难的技术。自从16世纪卡达诺关于三次方程的工作以来，人们已经知道，作为中介步骤，需要复数来寻找某些问题的真正解，即使这些数字既不出现在语句中，也不出现在问题本身的解中。因此，到上个世纪中叶，人们认识到复杂分析与数学、科学和技术的许多基本问题深深地交织在一起。其中最壮观的例子之一就是利用“狄利克莱原理”(它产生于流体力学)来理解单复变量函数理论的基础，进而在流体流动研究中使用复变分析。近年来，复分析的许多研究都集中在多个复变量的函数上，其中狄利克莱原理的对应出现在众所周知的霍奇理论和所谓的d-bar Neumann问题中。科恩正在做这方面的研究，这项研究还涉及偏微分方程、调和分析和几何方面的工作。科恩目前正在研究偏微分方程研究中出现的“光滑性”和“奇点传播”问题。

Proposal:DMS-9801626 Principal Investigator: Joseph J. Kohn Abstract: Kohn will continue to do research in the theory of several complex variables, partial differential equations and related topics. In particular, he is interested in: global regularity, hypoellipticity, Holder and L-p estimates, domains with piecewise smooth boundaries, and embeddings of CR manifolds. Kohn is currently working on the subject of hypoellipticity both for boundary behavior in several complex variables and for differential and pseudodifferential operators. His interest is in finding qualitative (be they geometric or algebraic) conditions under which hypoellipticity holds. The technical part of much of this research is connected with estimates which do not "gain" smoothness, so the error terms are of the same magnitude as the object one is trying to estimate. Kohn is developing techniques to deal with this type of difficulty. Since Cardano's work on the cubic equation in the sixteenth century it has been known that complex numbers are needed, as an intermediary step, to find real solutions of certain problems even when such numbers appear neither in the statements nor in the solutions of the problems themselves. Thus, by the middle of the last century, it was recognized that complex analysis is deeply intertwined with many fundamental problems of mathematics, science, and technology. One of the most spectacular instances of this is the use of "Dirichlet's principle" (which arises in fluid mechanics) to understand the foundations of the theory of functions of one complex variable and, in turn, the use of complex analysis in the study of fluid flows. In recent times much of the research in complex analysis has focused on functions of several complex variables, where the counterpart of the Dirichlet principle appears in what is known as Hodge theory and in the so-called d-bar Neumann problem. Kohn is doing research in this area, research that also involves work in partial differential equations, harmonic a nalysis, and geometry. Kohn is currently working on problems of "smoothness" and "propagation of singularities" that arise in the study of partial differential equations.