The central theme of this paper is the variational analysis of homeomorphisms $h: {mathbb X} overset{textnormal{tiny{onto}}}{longrightarrow} {mathbb Y}$ between two given domains ${mathbb X}, {mathbb Y} subset {mathbb R}^n$. The authors look for the extremal mappings in the Sobolev space ${mathscr W}^{1,n}({mathbb X},{mathbb Y})$ which minimize the energy integral ${mathscr E}_h=int_{{mathbb X}} ,|!|, Dh(x) ,|!|,^n, textrm{d}x$. 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.