Partial Differential Equations

本项目主要研究多复变理论中和CR流形上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，以及当次椭圆性失效时的次椭圆性.相关的问题是Hoelder连续性和解的真实的解析性。我们也将继续我们的研究整体正则性CR流形上使用microlocal方法.偏微分方程自然出现在物理，化学，工程，经济，和数学的几个不同的领域.有时这些方程可以显式求解，但更多的时候他们不能，在这种情况下，问题是要找到是否存在的解决方案，如果是这样，找到它的各种属性和近似.这是通过估算来实现的。在这里，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普遍性.我们使用的基本概念是“次椭圆乘子”，这是一个工具，它转换成代数和几何问题的估计搜索. 这个概念已被证明是非常有效的，不仅在proveingestimates，而且在各种代数和几何问题的研究。

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.