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The Cauchy-Riemann Complex

Integral Formulae and Neumann Problem

  • Book
  • © 2002

Overview

Authors:
  1. Ingo Lieb

    1. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

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  2. Joachim Michel

    1. Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Calais Cedex, France

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  • Advanced Methods of Complex Analysis Applied to Classical and New Problems

Part of the book series: Aspects of Mathematics (ASMA, volume 34)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-X
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  2. Introduction

    • Ingo Lieb, Joachim Michel
    Pages 1-7

  3. The Bochner-Martinelli-Koppelman Formula

    • Ingo Lieb, Joachim Michel
    Pages 9-57

  4. The Calculus of Cauchy-Fantappiè Forms

    • Ingo Lieb, Joachim Michel
    Pages 59-72

  5. Strictly Pseudoconvex Domains in ℂn

    • Ingo Lieb, Joachim Michel
    Pages 73-127

  6. Strictly Pseudoconvex Manifolds

    • Ingo Lieb, Joachim Michel
    Pages 129-181

  7. The \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafyOaIyRbae % baaaa!376B! \bar \partial \)-Neumann Problem

    • Ingo Lieb, Joachim Michel
    Pages 183-210

  8. Integral Representations for the \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafyOaIyRbae % baaaa!376B! \bar \partial \) -Neumann Problem

    • Ingo Lieb, Joachim Michel
    Pages 211-253

  9. Regularity Properties of Admissible Operators

    • Ingo Lieb, Joachim Michel
    Pages 255-299

  10. Regularity of the \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafyOaIyRbae % baaaa!376B! \bar \partial \)-Neumann Problem and Applications

    • Ingo Lieb, Joachim Michel
    Pages 301-335

  11. Back Matter

    Pages 337-362
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Keywords

About this book

This book presents complex analysis of several variables from the point of view of the Cauchy-Riemann equations and integral representations. A more detailed description of our methods and main results can be found in the introduction. Here we only make some remarks on our aims and on the required background knowledge. Integral representation methods serve a twofold purpose: 1° they yield regularity results not easily obtained by other methods and 2°, along the way, they lead to a fairly simple development of parts of the classical theory of several complex variables. We try to reach both aims. Thus, the first three to four chapters, if complemented by an elementary chapter on holomorphic functions, can be used by a lecturer as an introductory course to com­ plex analysis. They contain standard applications of the Bochner-Martinelli-Koppelman integral representation, a complete presentation of Cauchy-Fantappie forms giving also the numerical constants of the theory, and a direct study of the Cauchy-Riemann com­ plex on strictly pseudoconvex domains leading, among other things, to a rather elementary solution of Levi's problem in complex number space en. Chapter IV carries the theory from domains in en to strictly pseudoconvex subdomains of arbitrary - not necessarily Stein - manifolds. We develop this theory taking as a model classical Hodge theory on compact Riemannian manifolds; the relation between a parametrix for the real Laplacian and the generalised Bochner-Martinelli-Koppelman formula is crucial for the success of the method.

Authors and Affiliations

  • Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

    Ingo Lieb

  • Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Calais Cedex, France

    Joachim Michel

About the authors

Prof. Dr. Ingo Lieb ist Professor für Mathematik an der Universität Bonn. Er ist Autor der beiden Bücher "Funktionentheorie" und "Ausgewählte Kapitel aus der Funktionentheorie" in der Reihe vieweg studium/Aufbaukurs Mathematik.
Prof. Dr. Joachim Michel ist Professor für Mathematik am "Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville" (L.M.P.A.) in Calais, Frankreich.

Bibliographic Information

  • Book Title: The Cauchy-Riemann Complex

  • Book Subtitle: Integral Formulae and Neumann Problem

  • Authors: Ingo Lieb, Joachim Michel

  • Series Title: Aspects of Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-91608-2

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Fachmedien Wiesbaden 2002

  • Softcover ISBN: 978-3-322-91610-5Published: 27 July 2012

  • eBook ISBN: 978-3-322-91608-2Published: 06 December 2012

  • Series ISSN: 0179-2156

  • Edition Number: 1

  • Number of Pages: X, 362

  • Topics: Partial Differential Equations, Analysis

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