Mathematical Sciences: Partial Differential Equations

强调偏微分与 方程和多复变量理论将是 这是数学研究项目的主题。 工作将 研究了正则性和解的存在性， 微分方程，反映了边界的行为， 全纯映射的局部和整体嵌入定理 Cauchy-Riemann流形，全纯函数的研究 光滑边界值和超曲面的结构 双曲空间 这项工作，其中大部分仍在继续，导致 研究几何对象，如局部和全局不变量 流形的边界，叶理复品种和 全纯性的包络 目前最紧迫的问题是， d-杆Neumann问题解的保持器估计 有限型弱伪凸域及相应的 CR流形上相应算子的估计。 一 最近在这方面迈出了重要一步， 估计数是在两个复杂的方面确定的。 的 向更高维度的推广已经得到了相当大的 注意，但尚未出现突破。 工作将 首先考虑三维或多维的域 其Levi形式至多有一个退化特征值，然后 转移到那些形式是可对角化的。 另一条调查线将继续调查 涉及线性微分算子，具有简单和 双重特征。这项工作的目的是确定 超定系统的条件，这意味着 操作员是hypoelliptic。 术语的意思很简单， 任何非齐次方程的解至少是光滑的， 方程中的非齐次部分。

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.