Partial Differential Equations

本课题主要研究多复变理论和CR流形中出现的偏微分方程解的正则性。我们特别计划研究亚椭圆乘法器的各种性质，以及当亚椭圆失效时的亚椭圆性。相关的问题是霍尔德连续性和解的实解析性。我们还将继续使用微局部方法研究CR流形的全局正则性。偏微分方程在物理、化学、工程、经济学和数学的许多不同领域中自然出现。有时这些方程可以明确地解出来，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近似它。这是通过估算来实现的。这里我们研究某些特殊的基本偏微分方程的估计，但我们开发的技术适用于更大的一般性。我们使用的基本概念是“次椭圆乘法器”，这是一个将搜索估计转换为代数和几何问题的工具。这个概念不仅在证明估计，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.