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$n$-Harmonic Mappings between Annuli: The Art of Integrating Free Lagrangians
 
Tadeusz Iwaniec Syracuse University, Syracuse, NY and University of Helsinki, Helsinki, Finland
Jani Onninen Syracuse University, Syracuse, NY
$n$-Harmonic Mappings between Annuli
eBook ISBN:  978-0-8218-9008-0
Product Code:  MEMO/218/1023.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
$n$-Harmonic Mappings between Annuli
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$n$-Harmonic Mappings between Annuli: The Art of Integrating Free Lagrangians
Tadeusz Iwaniec Syracuse University, Syracuse, NY and University of Helsinki, Helsinki, Finland
Jani Onninen Syracuse University, Syracuse, NY
eBook ISBN:  978-0-8218-9008-0
Product Code:  MEMO/218/1023.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2182012; 105 pp
    MSC: Primary 30; 35;

    The central theme of this paper is the variational analysis of homeomorphisms \(h: {\mathbb X} \overset{\text{onto}}{\longrightarrow} {\mathbb Y}\) between two given domains \({\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n\). The authors look for the extremal mappings in the Sobolev space \({\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})\) which minimize the energy integral \[{\mathscr E}_h=\int_{{\mathbb X}} \| Dh(x) \|^n\, \mathrm{d}x\,.\] Because of the natural connections with quasiconformal mappings this \(n\)-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal \(n\)-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.

  • Table of Contents
     
     
    • Chapters
    • Preface
    • 1. Introduction and Overview
    • 1. Principal Radial $n$-Harmonics
    • 2. Nonexistence of $n$-Harmonic Homeomorphisms
    • 3. Generalized $n$-Harmonic Mappings
    • 4. Notation
    • 5. Radial $n$-Harmonics
    • 6. Vector Calculus on Annuli
    • 7. Free Lagrangians
    • 8. Some Estimates of Free Lagrangians
    • 9. Proof of Theorem
    • 2. The $n$-Harmonic Energy
    • 10. Harmonic Mappings between Planar Annuli, Proof of Theorem
    • 11. Contracting Pair, $ \mbox {Mod}\, {\mathbb A}^{\! \ast } \leqslant \mbox {Mod}\, {\mathbb A}$
    • 12. Expanding Pair, $\mbox {Mod}\, {\mathbb A}^{\! \ast } > \mbox {Mod}\, {\mathbb A}$
    • 13. The Uniqueness
    • 14. Above the Upper Nitsche Bound, $n \geqslant 4$
    • 15. Quasiconformal Mappings between Annuli
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2182012; 105 pp
MSC: Primary 30; 35;

The central theme of this paper is the variational analysis of homeomorphisms \(h: {\mathbb X} \overset{\text{onto}}{\longrightarrow} {\mathbb Y}\) between two given domains \({\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n\). The authors look for the extremal mappings in the Sobolev space \({\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})\) which minimize the energy integral \[{\mathscr E}_h=\int_{{\mathbb X}} \| Dh(x) \|^n\, \mathrm{d}x\,.\] Because of the natural connections with quasiconformal mappings this \(n\)-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal \(n\)-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.

  • Chapters
  • Preface
  • 1. Introduction and Overview
  • 1. Principal Radial $n$-Harmonics
  • 2. Nonexistence of $n$-Harmonic Homeomorphisms
  • 3. Generalized $n$-Harmonic Mappings
  • 4. Notation
  • 5. Radial $n$-Harmonics
  • 6. Vector Calculus on Annuli
  • 7. Free Lagrangians
  • 8. Some Estimates of Free Lagrangians
  • 9. Proof of Theorem
  • 2. The $n$-Harmonic Energy
  • 10. Harmonic Mappings between Planar Annuli, Proof of Theorem
  • 11. Contracting Pair, $ \mbox {Mod}\, {\mathbb A}^{\! \ast } \leqslant \mbox {Mod}\, {\mathbb A}$
  • 12. Expanding Pair, $\mbox {Mod}\, {\mathbb A}^{\! \ast } > \mbox {Mod}\, {\mathbb A}$
  • 13. The Uniqueness
  • 14. Above the Upper Nitsche Bound, $n \geqslant 4$
  • 15. Quasiconformal Mappings between Annuli
Review Copy – for publishers of book reviews
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Accessibility – to request an alternate format of an AMS title
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