Partial Differential Equations

Princeton University, Princeton NJ

Investigators

Abstract

This project is mainly concerned with the study of the regularity of solutions of partial differential equations that arise in the theory of several complex variables and on CR manifolds. In particular we plan to study various properties of subelliptic multipliers and also hypoellipticity when subellipticity fails. Related questions are Hoelder continuity and real analyticity of solutions. We also will continue our study 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 they cannot and, in that case, the problem is to find whether the solution exists and if so to find various of its properties and to approximate it. This is accomplished by means of estimates. Here we study estimates for certain special fundamental partial differential equations but the techniques we develop are applicable in much greater generality. The basic concept that we use are "subelliptic multipliers" this is a tool which converts the search for estimates into problems of algebra and geometry. This concept has proved very effective not only in proving estimates but also in the study of various algebraic and geometric problems.