Mathematical Sciences: Partial Differential Equations in Complex Analysis

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This project focuses on the interaction between studies of partial differential equations and the theory of several complex variables. The history of these interactions has been long and fruitful, with each subject providing valuable information to the other. The present work builds on a recent formulation of a unique continuation property for the inhomogeneous Cauchy-Riemann equations in several variables that will yield information about boundary behavior of holomorphic mappings between domains in complex space. Efforts will be made to verify the property in the most important categories of domains. Work will also be done using recent results about the regularity of the Cauchy-Riemann equations combined with geometric arguments to prove regularity theorems for the Bergman projection. Such results also have applications to mappings problems in several complex variables. A program to express the classical solutions to the Dirichlet and Neumann problems for the laplace operator on domains in the plane in terms of the Szego projection and kernel has just been completed. These results give rise to new and practical methods for numerically computing solutions to classical problems in partial differential equations and conformal mapping which are of interest in the applied sciences. Work will continue exploring applications of the Szego projection to related problems. Students on the project will be used to test the efficiency of the resulting numerical methods. The theory of several complex variables plays a central role in the analysis of linear partial differential equations. It provides an excellent environment in which tools from geometry, analysis and functional analysis can be brought to bear on equations which relate to modeling of the physical world.