eBook ISBN:  9780821890080 
Product Code:  MEMO/218/1023.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
eBook ISBN:  9780821890080 
Product Code:  MEMO/218/1023.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 218; 2012; 105 ppMSC: 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


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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