  We investigate problems in large-strain magnetoelasticity when a
Dzyaloshinskii-Moriya interaction term is included, too. While the
magnetoelastic stored energy density is described in the Lagrangean setting,
purely magnetic terms are considered in the Eulerian one. This requires careful
treatment of the invertibility of admissible elastic deformations and of the
regularity of the inverse maps. Besides the existence of a minimizer in the
static case, we also show that the model can be extended to an evolutionary
situation and enriched by a rate-independent dissipation. In this case, we
prove that an energetic solution exists.
