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$$n$$-Harmonic Mappings between Annuli: The Art of Integrating Free Lagrangians
Tadeusz Iwaniec, Syracuse University, NY, and University of Helsinki, Finland, and Jani Onninen, Syracuse University, NY
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Memoirs of the American Mathematical Society
2012; 105 pp; softcover
Volume: 218
ISBN-10: 0-8218-5357-0
ISBN-13: 978-0-8218-5357-3
List Price: US$70 Individual Members: US$42
Institutional Members: US\$56
Order Code: MEMO/218/1023

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.

• Introduction and overview
Part 1. Principal Radial $$n$$-Harmonics
• Nonexistence of $$n$$-Harmonic homeomorphisms
• Generalized $$n$$-Harmonic mappings
• Notation
• Radial $$n$$-harmonics
• Vector calculus on annuli
• Free Lagrangians
• Some estimates of free Lagrangians
• Proof of Theorem 1.15
Part 2. The $$n$$-Harmonic Energy
• Harmonic mappings between planar annuli, Proof of Theorem 1.8
• Contracting Pair, $$\mbox{Mod}\, {\mathbb A}^{\! \ast} \leqslant \mbox{Mod}\, {\mathbb A}$$
• Expanding Pair, $$\mbox{Mod}\, {\mathbb A}^{\! \ast} > \mbox{Mod}\, {\mathbb A}$$
• The uniqueness
• Above the upper Nitsche bound, $$n \geqslant 4$$
• Quasiconformal mappings between annuli
• Bibliography