Periodic homogenization of integral energies under space-dependent differential constraints

Abstract

A homogenization result for a family of oscillating integral energies

is presented, where the fields $u_{ϵ}$ are subjected to first order linear differential constraints depending on the space variable $x$. The work is based on the theory of $\mathcal{A}$-quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of $\mathcal{A}$-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.