n-Harmonic mappings between annuli

The central theme of this paper is the variational analysis of homeomorphisms $h\text{\hspace{0.17em}⁣}:\mathbb{X}\text{onto}\mathbb{Y}$ 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 $${\mathcal{E}}_h={\int }_{\mathbb{X}}\mathrm{\mid }\mathrm{\mid }Dh(x)\mathrm{\mid }{\mathrm{\mid }}^{n}dx\mathrm{.}$$ 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.