We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form \begin{ equation*} \begin{ aligned} u_\varepsilon\in L^p(\Omega;\mathbb{ R}^d) \mapsto \int_\Omega f_R\Big(x,\frac{ x}{ \varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{ aligned} \end{ equation*} where $u_\varepsilon$ is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, $f_R$, 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, $\mathcal{ A}$, that is, $\mathcal{ A} u _\varepsilon =0$. Our results provide a generalization of related ones in the literature concerning the ${ \rm \mathcal{ A} =curl } $ case to more general differential operators $\mathcal{ A}$ with constant coefficients, and without coercivity assumptions on the Lagrangian $f_R$.
Categories: math.AP