Partial Differential Equations

本课题主要研究在多复变理论和CR流形上出现的偏微分方程解的正则性。特别地，我们计划研究次椭圆乘子的各种性质以及次椭圆性失效时的次椭圆性。与此相关的问题是解的Hoelder连续性和实解析性。我们还将继续利用微局部方法研究CR流形上的全局正则性。偏微分方程组在物理、化学、工程、经济学和一些不同的数学领域中都是自然产生的。有时这些方程可以显式求解，但更多的时候它们不能，在这种情况下，问题是找出解是否存在，如果存在，找到它的各种性质并逼近它。这是通过估计的方式实现的。在这里，我们研究某些特殊基本偏微分方程解的估计，但我们发展的技巧适用于更一般的情况。我们使用的基本概念是“亚椭圆乘子”，这是一种将估计的搜索转化为代数和几何问题的工具。这一概念不仅在证明估计方面，而且在各种代数和几何问题的研究中都被证明是非常有效的。

This project is mainly concerned with the study of the regularity ofsolutions of partial differential equations that arise in the theory ofseveral complex variables and on CR manifolds. In particular we plan tostudy various properties of subelliptic multipliers and alsohypoellipticity when subellipticity fails. Related questions are Hoeldercontinuity and real analyticity of solutions. We also will continue ourstudy of global regularity on CR manifolds using microlocal methods.Partial differential equations arise naturally in physics, chemistry,engineering, economics, and in several diverse areas of mathematics.Sometimes these equations can be solved explicitly but more often theycannot and, in that case, the problem is to find whether the solutionexists and if so to find various of its properties and to approximateit. This is accomplished by means of estimates. Here we study estimatesfor certain special fundamental partial differential equations but thetechniques we develop are applicable in much greater generality. Thebasic concept that we use are "subelliptic multipliers" this is a toolwhich converts the search for estimates into problems of algebra andgeometry. This concept has proved very effective not only in provingestimates but also in the study of various algebraic and geometricproblems.