  We investigate variational problems in large-strain magnetoelasticity, both
in the static and in the quasistatic setting. The model contemplates a mixed
Eulerian-Lagrangian formulation: while deformations are defined on the
reference configuration, magnetizations are defined on the deformed set in the
actual space. In the static setting, we establish the existence of minimizers.
In particular, we provide a compactness result for sequences of admissible
states with equi-bounded energies which gives the convergence of the
composition of magnetizations with deformations. In the quasistatic setting, we
consider a notion of dissipation which is frame-indifferent and we show that
the incremental minimization problem is solvable. Then, we propose a
regularization of the model in the spirit of gradient polyconvexity and we
prove the existence of energetic solutions for the regularized model.
