n-Harmonic mappings between annuli

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 ${\mathcal{W}}^{1,n}(\mathbb{X},\mathbb{Y})$ 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.

120 pages, 22 figures