| eBook ISBN: | 978-0-8218-9008-0 |
| Product Code: | MEMO/218/1023.E |
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| AMS Member Price: | $42.00 |
| 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 |
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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.
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Table of Contents
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Chapters
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Preface
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1. Introduction and Overview
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1. Principal Radial $n$-Harmonics
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2. Nonexistence of $n$-Harmonic Homeomorphisms
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3. Generalized $n$-Harmonic Mappings
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4. Notation
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5. Radial $n$-Harmonics
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6. Vector Calculus on Annuli
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7. Free Lagrangians
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8. Some Estimates of Free Lagrangians
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9. Proof of Theorem
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2. The $n$-Harmonic Energy
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10. Harmonic Mappings between Planar Annuli, Proof of Theorem
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11. Contracting Pair, $ \mbox {Mod}\, {\mathbb A}^{\! \ast } \leqslant \mbox {Mod}\, {\mathbb A}$
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12. Expanding Pair, $\mbox {Mod}\, {\mathbb A}^{\! \ast } > \mbox {Mod}\, {\mathbb A}$
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13. The Uniqueness
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14. Above the Upper Nitsche Bound, $n \geqslant 4$
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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
