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Lectures on the L2\mathcal{L}^2-Sobolev Theory of the \overline{\partial}-Neumann problem

  • Emil J. Straube

    Texas A&M University, College Station, USA
Lectures on the ℒ²-Sobolev Theory of the ∂-Neumann problem cover
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This book provides a thorough and self-contained introduction to the \overline{\partial}-Neumann problem, leading up to current research, in the context of the L2\mathcal{L}^2-Sobolev theory on bounded pseudoconvex domains in Cn\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 L2\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 L2\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.