  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.
