Partial Differential Equations and Complex Analysis

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

DMS - 0300408 PI: J McNeal Inst: Ohio State ABSTRACT In this project, I will work on four basic problems in several complex variables: A). The continuous extension of biholomorphic mappings between general smoothly bounded, pseudoconvex domains, in higher dimensional complex space, to their boundaries; B). The square-integrable cohomology of the Cauchy-Riemann operators in the Bergman and other Kaehler metrics; C). Hoelder regularity of solution operators to the Cauchy-Riemann equations; D). Potential-theoretic conditions related to compactness of the Neumann operator associated to the Cauchy-Riemann complex. Our attack on these problems will be through certain partial differential equations associated to the Cauchy-Riemann and Laplace operators. In our problems, we are in a situation where classical estimates do not provide enough information on the Cauchy-Riemann and Laplace operators to conclude the results we expect. Thus, our approach will be to perturb the Cauchy-Riemann and Laplace operators suitably, obtain strong estimates on the perturbed operators, and then connect these estimates to the unperturbed operators. The starting point is a twisted version of the Cauchy-Riemann complex.This twisted complex has already produced some powerful results in complex analysis and it is currently being extended to other contexts by several mathematicians, including the proposer. We view its continued development as one of the very fertile ideas presently ongoing in several complex variables. We also currently have three graduate students who are working on problems closely related to these topics; their problems deal with estimating the weighted Bergman projection on some finite type domains, studying the condition of self-bounded gradient and its relationship to subellipticity of the Neumann problem and hyperbolicity, and strengthening the Sobolev embedding theorem on convex domains of finite type. Our proposed research will make a significant contribution to certain broad,general question: by how much may one perturb a system of partial differential equations and still obtain information on the unperturbed system from the perturbed system? The results we expect will have many connections to areas of mathematics outside complex analysis, especially topology (through the connection between square-integrable cohomology and ordinary homology), differential equations,and algebraic geometry (by deepening our understanding of order of contact between algebraic and analytic varieties).In addition, however, since partial differential equations are ubiquitous throughout science and engineering, our perturbation methods will be useful outside pure mathematics. We believe that adaptations of our perturbation methods will apply to partial differential equations which arise in parts of current scientific theory and our proposed research will thereby strengthen ties and establish new connections between complex analysis and other fields of science.