Variational convergences under moving anisotropies

We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincaré inequalities and compactness properties. We prove several $\mathrm{\Gamma }$-convergence results, and apply the latter to the study of $H$-convergence of anisotropic linear differential operators.