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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

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
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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.

Table of Contents

  • 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
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