Homogenization of integral energies under periodically oscillating differential constraints (Q2628976)

The authors study integral representations for limits of oscillating integral energies \(u_\varepsilon\mapsto \int_\Omega f\left(x,\frac x{\varepsilon^\alpha},u_\varepsilon(x)\right)\,dx,\)\ where \(\Omega\subset\mathbb{R}^N\) is an open domain, \(\varepsilon\to 0^+\), and the fields \(u_\varepsilon\in L^p(\Omega;\mathbb{R}^d)\) are subjected to oscillating differential constraints in divergence form: \[ \mathcal{A}_\varepsilon^{\text{div}} u_\varepsilon:= \sum_{i=1}^N\frac \partial{\partial x_i}\left(A^i\left(\frac x{\varepsilon^\beta}\right)\right)\to 0\;\;\text{strongly in} \;W^{-1,p}(\Omega;\mathbb{R}^\ell),\;1<p<+\infty. \] The notion of \(\mathcal{A}\)-quasiconvexity and a two-scale convergence technique play the key role in their analysis.