The aim of the paper is to investigate the harmonic diffeomorphisms of the unit ball
onto a Riemannian manifold. It is shown that for
these diffeomorphisms are minimizing with respect to the energy functional over
mappings with trace condition. For
it is given a sufficient condition for this property in terms of the metric tensor. The case of finitely many punctual singularities is also treated. Interesting applications, especially for
, are indicated when the harmonic diffeomorphisms are SO(n)-equivariant in an appropriate sense. One answers some questions raised in papers of Gulliver, White, Jäger, Kaul, Baldes and Helein.