Mathematical Sciences: Partial Differential Equations

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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.