Partial Differential Equations in Several Complex Variables

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Mei-Chi Shaw intends to continue her investigation of partial Differential equations, which arise from problems in several complex variables. The problems discussed in this proposal include: the Cauchy-Riemann equations and the tangential Cauchy-Riemann complex on Non-smooth domains, the geometric aspects of these equations with curvature terms, and their relations with function theory on complex manifolds. New approaches to study these problems are described. Estimates for the tangential Cauchy-Riemann equations are closely related to the embedding theorems of abstract structures and the construction of bounded holomorphic functions, two of the most challenging problems in several complex variables and complex geometry.The intellectual merit of the proposed activity stems from the fact that these problems are at the forefront of several complex variables and partial differential equations. Their solution will advance all these intricately related fields and lead to significant progress in complex geometry and several complex variables. The P.I. will investigate the problems using new techniques and in collaboration with mathematicians from different areas. It will bring new perspective and greater depth to each field. These problems also have applications in applied mathematics and physics. The Hodge theorem on domains with corners and wedges has applications in electrokinetics and engineering. Its origin goes back to the classical Dirichlet Principle, the canonical solution to the energy minimizing problem. The Hodge theorem has developed into a beautiful theory and becomes an indispensable tool to many problems in partial differential equations, complex analysis, differential geometry and harmonic analysis. The proposed project is to advance our knowledge in this direction.