Periodic homogenization of integral energies under space-dependent differential constraints
Summary: A homogenization result for a family of oscillating integral energies \[ u_{\varepsilon}\mapsto\int_{\Omega}f\left(x,\frac x\varepsilon,u_\varepsilon(x)\right)dx,\quad\varepsilon\to0^+ \] is presented, where the fields \(u_{\varepsilon}\) 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.
scientific article; zbMATH DE number 6669854