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The central theme of this paper is the variational analysis of homeomorphisms h: X onto −→ Y between two given domains X,Y ⊂ Rn. We look for the extremal mappings in the Sobolev space W1,n(X,Y) which minimize the energy integral Eh =ZX ||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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