Mei-Chi Shaw will investigate topics in partial differential equations which arise from function theory in several complex variables. Problems in three specific areas are discussed in this proposal : the Cauchy-Riemann equations on nonsmooth domains, tangential Cauchy-Riemann equations and their interplay with singular integrals and geometric measure theory. Aspects of the Cauchy-Riemann equations on Lipschitz domains addressed include the complex Neumann boundary value problem and estimates of the Cauchy-Riemann equations. Function theory will be analyzed on strongly pseudoconvex Lipschitz domains, as well as the Bergman projection and biholomorphic maps. Recent results in harmonic analysis on Lipschitz domains are the main tools. Research on the regularity property of the global and local solutions of the tangential Cauchy-Riemann equation will be continued. This includes the existence and regularity theorems on CR manifolds which are Lipschitz or of higher codimension. Homotopy formulas, Szego projection, Hodge theory and the embeddings of abstract CR manifolds are also studied. Singular integral theory and geometric measure theory on Lipschitz curves and nonsmooth domains will play a major role in the approach to these problems.
These problems are at the interface of harmonic analysis, geometric measure theory, complex geometry and partial differential equations with rough coefficients, important fields all in modern analysis. Classically, harmonic analysis offers a powerful tool to solve partial differential equations, especially the Dirichlet and Neumann boundary value problems. Harmonic analysis is also intertwined with one complex variable, especially for function theory in the unit disc. In its modern version, harmonic analysis has evolved into singular integral theory and geometric measure theory. The development of one influences the other and both collectively are viewed as one of the most important and elegant theories in mathematics. In several complex variables, these overlapping areas have produced even richer and more profound results whose impact have shaped modern partial differential equations, several complex variables and harmonic analysis. Their influence even extends beyond these fields into other areas, like complex differential geometry, algebraic geometry and mathematical physics. While great progress has been made in the past few decades for the case when the domains are smooth, little is known when the domain is less regular. The investigation of these problems on nonsmooth domains is already central to the study of classical Dirichlet and Neumann boundary value problems using harmonic analysis. Built on these solid foundations, the PI intends to tackle more challenging problems in several complex variables. Their solution will advance all the aforementioned intricately related fields and open up a vast unexplored area.
