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n-Harmonic mappings between annuli

Authors
  • Iwaniec, Tadeusz
  • Onninen, Jani
Type
Preprint
Publication Date
Feb 04, 2011
Submission Date
Feb 04, 2011
Source
arXiv
License
Yellow
External links

Abstract

The central theme of this paper is the variational analysis of homeomorphisms $h\colon \mathbb X \onto \mathbb Y$ 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 \[ \mathscr E_h=\int_{\mathbb X} ||Dh(x)||^n dx. \] 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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