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We consider a homogenization problem associated with quasi-crystallinemultiple integrals of the form \begin{equation*} \begin{aligned}u_\varepsilon\in L^p(\Omega;\mathbb{R}^d) \mapsto \int_\Omegaf_R\Big(x,\frac{x}{\varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{aligned}\end{equation*} where $u_\varepsilon$ is subject to constant-coefficient linearpartial differential constraints. The quasi-crystalline structure of theunderlying composite is encoded in the dependence on the second variable of theLagrangian, $f_R$, and is modeled via the cut-and-project scheme thatinterprets the heterogeneous microstructure to be homogenized as an irrationalsubspace of a higher-dimensional space. A key step in our analysis is thecharacterization of the quasi-crystalline two-scale limits of sequences of thevector fields $u_\varepsilon$ that are in the kernel of a givenconstant-coefficient linear partial differential operator, $\mathcal{A}$, thatis, $\mathcal{A} u _\varepsilon =0$. Our results provide a generalization ofrelated ones in the literature concerning the ${\rm \mathcal{A} =curl } $ caseto more general differential operators $\mathcal{A}$ with constantcoefficients, and without coercivity assumptions on the Lagrangian $f_R$.

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