Relaxation in magnetostriction (Q1964687)

The author carries out the relaxation process for the class of nonlocal functionals defined as \[ I(m)=\int_\Omega \phi(x,u(x),m(x)) dx+ \int_\Omega\psi(x,u(x),\nabla u(x)) dx, \] where \(\Omega\) is an open subset of \({\mathbb R}^N\), \(m\in L^2(\Omega)\) and \(u\in H^1_0(\Omega)\) are coupled by the nonlocal differential constraint \[ \text{div}(-\nabla u+m)=0. \] The relaxed functional is computed in terms of Young measures associated with minimizing sequences of \(I\). In dimension two the magnetostriction functional is discussed in detail, and under some simplifying assumptions.