Ever since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis.
The book is an excellent introduction to a variety of topics and presents many of the basic elements of linear partial differential equations in the context of how they are applied to the study of complex analysis. The author treats the Dirichlet and Neumann problems for elliptic equations and the related Schauder regularity theory, and examines how those results apply to the boundary regularity of biholomorphic mappings. He studies the ?-Neumann problem, then considers applications to the complex function theory of several variables and to the Bergman projection.
The Dirichlet Problem in the Complex Plane
Review of Fourier Analysis
Elliptic Boundary Value Problems
A Degenerate Elliptic Boundary Value Problem
The ?- Neumann Problem
Applications of the ?- Neumann Problem
The Local Solvability Issue and a Look Back.
"This well-written book is a valuable contribution to the broad field of interactions between complex analysis and partial differential equations...Moreover, the book can be used for individual studies, because fundamental concepts and important theorems are explained in detail."
-Mathematical Reviews, Issue 94a