On Uniqueness in the Affine Boundary Value Problem of the Nonlinear Elastic Dielectric

An integral identity is constructed from properties of the energy momentum tensor and is used to demonstrate uniqueness of the displacement on star-shaped regions to the affine boundary value problem of the nonlinear homogeneous elastic dielectric. The method of proof, nontrivially adapted from that of the corresponding elastic problem, assumes the electric enthalpy function to be rank-one convex and strictly quasiconvex. Furthermore, for a given displacement gradient, the electric quantities are proved unique for specified nonaffine and nonuniform electric boundary conditions subject to the electric enthalpy and strain energy functions satisfying additional convexity conditions.