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We analyze the asymptotic behavior of a multiscale problem given by asequence of integral functionals subject to differential constraints conveyedby a constant-rank operator with two characteristic length scales, namely thefilm thickness and the period of oscillating microstructures, by means of$\Gamma$-convergence. On a technical level, this requires a subtile merging ofhomogenization tools, such as multiscale convergence methods, with dimensionreduction techniques for functionals subject to differential constraints. Oneobserves that the results depend critically on the relative magnitude betweenthe two scales. Interestingly, this even regards the fundamental question oflocality of the limit model, and, in particular, leads to new findings also inthe gradient case.

Heterogeneous thin films: Combining homogenization and dimension reduction with directors