  This paper concerns the rigorous periodic homogenization for a weakly coupled
electroelastic system of a nonlinear electrostatic equation with an elastic
equation enriched with electrostriction. Such coupling is employed to describe
dielectric elastomers or deformable (elastic) dielectrics. It is shown that the
effective response of the system consists of a homogeneous dielectric elastomer
described by a nonlinear weakly coupled system of PDEs whose coefficients
depend on the coefficients of the original heterogeneous material, the geometry
of the composite and the periodicity of the original microstructure. The
approach developed here for this nonlinear problem allows obtaining an explicit
corrector result for the homogenization of monotone operators with minimal
regularity assumptions. Two $L^p-$gradient estimates for elastic systems with
discontinuous coefficients are also obtained.
