# Lectures on the \(\mathcal{L}^{2}\)-Sobolev Theory of the \(∂\)-Neumann Problem

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*Emil J. Straube*

A publication of the European Mathematical Society

This book provides a thorough and self-contained introduction to the
\(\bar{\partial}\)-Neumann problem, leading up to current research, in
the context of the \(\mathcal{L}^{2}\)-Sobolev theory on bounded
pseudoconvex domains in \(\mathbb{C}^{n}\). It grew out of courses for
advanced graduate students and young researchers given by the author at the
Erwin Schrödinger International Institute for Mathematical Physics and at
Texas A & M University.

The introductory chapter provides an overview of the contents and puts them
in historical perspective. The second chapter presents the basic
\(\mathcal{L}^{2}\)-theory. Following is a chapter on the subelliptic
estimates on strictly pseudoconvex domains. The two final chapters on
compactness and on regularity in Sobolev spaces bring the reader to the
frontiers of research.

Prerequisites are a solid background in basic complex and functional
analysis, including the elementary \(\mathcal{L}^{2}\)-Sobolev theory
and distributions. Some knowledge in several complex variables is helpful.
Concerning partial differential equations, not much is assumed. The elliptic
regularity of the Dirichlet problem for the Laplacian is quoted a few times,
but the ellipticity results needed for elliptic regularization in the third
chapter are proved from scratch.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in \(\mathcal{L}^{2}\)-Sobolev Theory.