Homogenization of Quasi-Crystalline Functionals via Two-Scale-Cut-and-Project Convergence
Rita Ferreira, Irene Fonseca, Raghavendra Venkatraman · SIAM Journal on Mathematical Analysis · 2021
We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form $u_\varepsilon\in L^p(\Omega;\mathbbm{R}^d) \mapsto \int_\Omega f_R(x,\frac{x}{\varepsilon}, u_\varepsilon(x)), dx,$ where \(u_ǎrepsilon) 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_ǎrepsilon) that are in the kernel of a given constant-coefficient linear partial differential operator, (\mathcalA), that is, (\mathcalA u _ǎrepsilon =0). Our results provide a generalization of related ones in the literature concerning the (\mathcalA =curl ) case to more general differential operators (\mathcalA) with constant coefficients and without coercivity assumptions on the Lagrangian (f_R).