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Partial Differential Equations in Several Complex Variables
So-Chin Chen, National Tsing-Hua University, Hsinchu, Taiwan, and Mei-Chi Shaw, University of Notre Dame, IN
A co-publication of the AMS and International Press.
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AMS/IP Studies in Advanced Mathematics
2001; 380 pp; softcover
Volume: 19
ISBN-10: 0-8218-2961-0
ISBN-13: 978-0-8218-2961-5
List Price: US$57
Member Price: US$45.60
Order Code: AMSIP/19.S
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This book is intended both as an introductory text and as a reference book for those interested in studying several complex variables in the context of partial differential equations. In the last few decades, significant progress has been made in the fields of Cauchy-Riemann and tangential Cauchy-Riemann operators. This book gives an up-to-date account of the theories for these equations and their applications.

The background material in several complex variables is developed in the first three chapters, leading to the Levi problem. The next three chapters are devoted to the solvability and regularity of the Cauchy-Riemann equations using Hilbert space techniques. The authors provide a systematic study of the Cauchy-Riemann equations and the \(\bar\partial\)-Neumann problem, including \(L^2\) existence theorems on pseudoconvex domains, \(\frac 12\)-subelliptic estimates for the \(\bar\partial\)-Neumann problems on strongly pseudoconvex domains, global regularity of \(\bar\partial\) on more general pseudoconvex domains, boundary regularity of biholomorphic mappings, irregularity of the Bergman projection on worm domains.

The second part of the book gives a comprehensive study of the tangential Cauchy-Riemann equations. Chapter 7 introduces the tangential Cauchy-Riemann complex and the Lewy equation. An extensive account of the \(L^2\) theory for \(\square_b\) and \(\bar\partial_b\) is given in Chapters 8 and 9. Explicit integral solution representations are constructed both on the Heisenberg groups and on strictly convex boundaries with estimates in Hölder and \(L^p\) spaces. Embeddability of abstract \(CR\) structures is discussed in detail in the last chapter.

This self-contained book provides a much-needed introductory text to several complex variables and partial differential equations. It is also a rich source of information to experts.

Titles in this series are co-published with International Press, Cambridge, MA.

Readership

Graduate students and research mathematicians interested in several complex variables and PDEs.

Reviews

"Anyone planning to do research in this area will want to have a copy of the book."

-- Mathematical Reviews

Table of Contents

  • Real and complex manifolds
  • The Cauchy integral formula and its applications
  • Holomorphic extension and pseudoconvexity
  • \(L^2\) theory for \(\overline\partial\) on pseudoconvex domains
  • The \(\overline\partial\)-Neumann problem on strongly pseudoconvex manifolds
  • Boundary regularity for \(\overline\partial\) on pseudoconvex domains
  • Cauchy-Riemann manifolds and the tangential Cauchy-Riemann complex
  • Subelliptic estimates for second order differential equations and \(\square_b\)
  • The tangential Cauchy-Riemann complex on pseudoconvex \(CR\) manifolds
  • Fundamental solutions for \(\square_b\) on the Heisenberg group
  • Integral representations for \(\overline\partial\) and \(\overline\partial_b\)
  • Embeddability of abstract \(CR\) structures
  • Appendix
  • Bibliography
  • Table of notation
  • Index
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