This project addresses fundamental questions at the intersection of several complex variables and partial differential equations. Two thrusts concern global regularity and compactness of the d-bar-Neumann operator in a domain. The principal investigator has recently developed a theory that unifies the known positive results concerning global regularity that is expected to provide an important step towards a desired characterization of this property in terms of properties of the boundary of the domain. It is not known how far the known potential theoretic sufficient conditions for compactness of the d-bar-Neumann operator are from being necessary. Building on his recent work with a graduate student (S. Munasinghe) that provides a different approach, the investigator will seek to characterize compactness. A third thrust of the project initiates a new direction of research: the available evidence suggests that regularity properties of the d-bar-Neumann operator in a domain should have positive implications for the existence of a Stein neighborhood basis of the closure of the domain. The fourth thrust results from a collaboration of the investigator with the postdoctoral researcher A. Raich, in which he and the principal investigator showed that the classical sufficient conditions for compactness of the operators in the interior of a domain (with a necessary modification) also yield compactness of the operators on the boundary. The goal is to combine their methods with earlier methods of the investigator to show that finite type implies subellipticity on the boundary.

The study of analysis in several complex variables is motivated both by the centrality of the subject to mathematics and by its usefulness. For example, one of the central laws of nature, causality, when transcribed via a mathematical device called the Fourier transform, leads immediately to analytic functions of several (in this case four) complex variables. Partial differential equations, on the other hand, arise in all areas of science that deal with systems that change over time: physics, engineering, economics, biology, meteorology, environmental sciences, and others. This project is thus located on the basic research end at the intersection of two areas of mathematics that are central to the scientific and technological enterprise. It will impact human resources development directly through funding for graduate students and through the investigator's supervision of postdoctoral researchers, and indirectly through the investigator's organization of and participation in workshops/conferences and through his expository writing.

Project Report

This project is located at the basic research end at the intersection of Several Complex Variables and Partial Differential Equations. These are two areas of mathematics that are central to the scientific and technological enterprise, needed for important applications in such fields as physics, engineering, economics, biology, meteorology, environmental sciences, and others. It impacts human resource development directly through the PI's supervision of graduate students and post-doctoral researchers, and indirectly through his expository writing. One of the central problems in the area described above is the so called d-bar-Neumann problem. This consists of an overdetermined system of partial differential equations, with boundary conditions. Roughly speaking, one would like to know that the solution is as smooth (i.e. as good) as the given data. Problems occur at the boundary of the region where one considers the equation(s), because the boundary conditions are in some sense "degenerate" (i.e. bad). This project deals with specific problems that arise from this circle of ideas, and some related questions. The results obtained add significantly to the existing knowledge base. Specifically, there are five journal publications that appeared or were accepted for publication during the project period, one research monograph, and one conference proceedings that the PI co-edited. Among the co-authors are two former students of the PI and one former post-doc. In addition, two graduate students started their thesis research under the PI's supervision.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Bruce P. Palka
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Texas A&M Research Foundation
College Station
United States
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