Relationship between existence of energy minimizers of incompressible and nearly incompressible magnetostrictive materials

Models of incompressible and slightly compressible magnetostrictive materials are introduced. They are given by the free energy functionals which depend on magnetization and elastic deformation as well as on their gradients. We demonstrate the existence of minimum of an energy functional for a slightly compressible material. We also prove a theorem on convergence of a sequence of minimizers of less and less compressible material energy functionals to a minimizer of energy of incompressible material. Besides the existence of solution of the incompressible magnetostrictive problem is obtained.