Periodic homogenization of integral energies under space-dependent differential constraints

Abstract

A homogenization result for a family of oscillating integral energies u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+ is presented, where the fields u_{\epsilon} are subjected to first order linear differential constraints depending on the space variable x . The work is based on the theory of \mathscr 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 \mathscr A -quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.