Explicit corrector in homogenization of monotone operators and its application to nonlinear dielectric elastomer composites

Abstract

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.

We provide a new proof to extend the explicit first-order corrector result in the first version of this paper. The new explicit corrector result (cf. Theorem 1) holds globally and unifies the previous classical correct results in homogenization of the divergence equation (both linear and nonlinear). New references are added. Comments are welcome!