Existence of minimizers foe a variational problem in two-dimensional nonlinear magnetoelasticity

We prove the existence of energy-minimizing configurations for a two-dimensional, variational model of magnetoelastic materials capable of large deformations. The model is based on an energy functional which is the sum of the nonlocal self-energy (the energy stored in the magnetic field generated by the body, and permeating the whole ambient space) and of the local anisotropy energy, which is not weakly lower semicontinuous. A further feature of the model is the presence of a non-convex constraint on one of the unknowns, the magnetization, which is a unit vector field