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$n$-Harmonic Mappings Between Annuli: The Art of Integrating Free Lagrangians

About this Title

Tadeusz Iwaniec, Department of Mathematics, Syracuse University, Syracuse, New York 13244 and Department of Mathematics and Statistics, University of Helsinki, Finland and Jani Onninen, Department of Mathematics, Syracuse University, Syracuse, New York 13244

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 218, Number 1023
ISBNs: 978-0-8218-5357-3 (print); 978-0-8218-9008-0 (online)
Published electronically: September 19, 2011
Keywords:$n$-Harmonics, Extremal problems, Quasiconformal mappings, Variational integrals
MSC: Primary 30C65, 30C75, 35J20

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Table of Contents


  • Preface
  • Chapter 1. Introduction and overview
  • Part 1. Principal radial $n$-harmonics
  • Part 2. The $n$-harmonic energy


The central theme of this paper is the variational analysis of homeomorphisms $h \colon \mathbb{X} \overset{\text{onto}}{\longrightarrow} \mathbb{Y}$ between two given domains $\mathbb{X} , \mathbb{Y} \subset \mathbb{R}^n$. We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb{X},\mathbb{Y})$ which minimize the energy integral

$\displaystyle {\mathscr E}_h=\int_{{\mathbb{X}}} \vert\vert Dh(x) \vert\vert ^n \textrm{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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