Nonlocal Variational Problems in Nonlinear Electromagneto-Elastostatics

Abstract

The effects of arbitrary applied electric and magnetic fields on unshielded, nonlinear, deformable, polarizable, magnetizable, nonconducting bodies are studied. Both monotone materials (dielectric, paramagnetic, etc.) and classical ferromagnetic materials are considered. The lack of shielding forces us to consider unknown fields outside of the material. This leads to nonlocal (“shape-dependent”) effects. The work of Ball is extended [“Convexity conditions and existence theorems in nonlinear elasticity,” Arch. Rat. Mech. Anal., 63 (1977), pp. 337–403] to get an existence theory using direct methods of the calculus of variations.