Mathematical Sciences: Partial Differential Equations

强调偏微分方程和多复变量理论之间的相互作用将是这个数学研究项目的主要主题。我们将研究反映全纯映象边界行为的微分方程解的正则性和存在性，Cauchy-Riemann流形的局部和整体嵌入定理，具有光滑边值的全纯函数的研究，以及双曲空间中超曲面的结构。这项工作，其中大部分仍在继续，导致几何对象的研究，如具有边界的流形的局部和全局不变量，复变簇的叶层和全纯包络。此时最紧迫的问题是获得有限类型弱伪凸域上d-bar Neumann问题解的Holder估计以及相应的CR流形上相应算子的估计。最近在这方面迈出了重要的一步，在两个复杂的方面确定了估计数。推广到更高的维度已经引起了相当大的关注，但尚未出现突破。工作将继续进行，首先考虑三维或更多维的区域，其Levi形式至多有一个简并的特征值，然后转移到形式可对角化的区域。另一项研究将涉及具有简单和双重特征的线性微分算子。这项工作的目的是确定超定系统上的条件，这些条件意味着算子是亚椭圆的。这个术语的简单意思是，任何非齐次方程的解至少与方程的非齐次部分的解一样光滑。

Emphasis on the interplay between partial differential equations and the theory of several complex variables will be the main theme of this mathematical research project. Work will be done on the study of regularity and existence of solutions of differential equations which reflect the boundary behavior of holomorphic mappings, on local and global embedding theorems for Cauchy-Riemann manifolds, studies of holomorphic functions with smooth boundary values and on the structure of hypersurfaces in hyperbolic space. This work, much of it continuing, leads to the study of geometrical objects such as local and global invariants of manifolds with boundary, foliations by complex varieties and envelopes of holomorphy. The most pressing problem at this time is that of obtaining Holder estimates for solutions of the d-bar Neumann problem for weakly pseudoconvex domains of finite type and the corresponding estimates for the corresponding operator on CR manifolds. A major step was taken in this direction recently when the estimates were established in two complex dimensions. The generalization to higher dimensions has received considerable attention, but no breakthrough has yet appeared. Work will proceed by first considering domains in three or more dimensions whose Levi form has at most one degenerate eigenvalue and then moving to those where the form is diagonalizable. Another line of investigation will pursue questions concerned with linear differential operators having simple and double characteristics. The object of this work is to determine conditions on overdetermined systems which imply that the operator is hypoelliptic. The terminology means simply that the solution of any inhomogeneous equation is at least as smooth as that of the inhomogeneous part of the equation.