Homogenization of quasi-crystalline functionals via two-scale-cut-and-project convergence

We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form {equation presented} dx, where uϵ is subject to constantcoefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, fR, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields u\varepsilon that are in the kernel of a given constant-coefficient linear partial differential operator, A, that is, A uϵ = 0. Our results provide a generalization of related ones in the literature concerning the A = curl case to more general differential operators A with constant coefficients and without coercivity assumptions on the Lagrangian fR.