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

TL;DR

This paper reviews recent advances in the $L^{2}$-Sobolev theory of the $ar{ ext{ extbeta}}$-Neumann problem, focusing on compactness and regularity issues in complex analysis boundary problems.

Contribution

It discusses recent developments in understanding the $L^{2}$-Sobolev regularity and compactness properties of the $ar{ ext{ extbeta}}$-Neumann problem on pseudoconvex domains.

Findings

Advances in compactness criteria for the $ar{ ext{ extbeta}}$-Neumann operator

Conditions for global regularity on pseudoconvex domains

Identification of cases where regularity fails

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.