Mathematical Sciences: Partial Differential Equations in Complex Analysis

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Work planned on this project centers on a single broad question: isolate the local and global boundary regularity properties of partial differential operators on domains in complex n-dimensional space. The research combines analytic methods of differential equations with geometric concepts in several complex variables. The focus will be on differential operators which play a natural role in the study of holomorphic functions, the so-called d-bar and box-b operators. Aside from intrinsic interest, the research has its roots in a fundamental discovery of Charles Fefferman in the 1970's that biholomorphic mappings between certain domains (pseudoconvex) have natural and continuous extensions to the respective boundaries. To achieve this, an integral transform, the Bergman operator must be shown to be analytic on the boundary of each domain. This research seeks to expand the results to a much larger class of domains known as weakly pseudoconvex. One must begin with the case of domains with real analytic boundaries. The necessary estimates have been obtained to yield positive results in the case of certain weakly pseudoconvex domains, in particular domains embedded in larger, biholomorphic domains can be treated. New local results suggest that global analytic regularity up to the boundary is true and accessible. Work will concentrate on those domains which are nonembeddable, for if the conjecture fails, it must do so in one of these circumstances. Applications of this work are planned in the general classification of domains and in the development of a priori estimates of solutions of partial differential equations.