Aspects of the $L^2$-Sobolev theory of the $\bar{\partial }$-Neumann problem

Abstract

The $\bar{\partial }$-Neumann problem is the fundamental boundary value problem in several complex variables. It features an elliptic operator coupled with non-coercive boundary conditions. The problem is globally regular on many, but not all, pseudoconvex domains. We discuss several recent developments in the $L^2$-Sobolev theory of the $\bar{\partial }$-Neumann problem that concern compactness and global regularity.