Partial Differential Equations

该项目主要研究多复变量理论和CR流形上的偏微分方程解的正则性。 特别是，我们计划研究亚椭圆乘法器的各种性质以及当亚椭圆性失效时的亚椭圆性。相关问题是霍尔德连续性和解的实分析性。我们还将继续使用微局域方法研究 CR 流形上的全局正则性。偏微分方程自然出现在物理、化学、工程、经济学和数学的几个不同领域中。有时这些方程可以显式求解，但更多时候不能，在这种情况下，问题是找出解是否存在，如果存在，则找到其各种属性并对其进行近似。这是通过估计来完成的。在这里，我们研究某些特殊基本偏微分方程的估计，但我们开发的技术具有更大的通用性。我们使用的基本概念是“次椭圆乘法器”，这是一种将估计搜索转换为代数和几何问题的工具。 事实证明，这个概念不仅在证明估计方面非常有效，而且在各种代数和几何问题的研究中也非常有效。

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.