Heterogeneous thin films: combining homogenization and dimension reduction with directors (Q2790395)

The authors examine the thin-film limit as \(\varepsilon\downarrow 0\) for the variational principles NEWLINE\[NEWLINEG_\varepsilon [v]\to \min, v:\Omega_\varepsilon\to \mathbb{R}^m, \mathcal{A}v=0\text{ in }\Omega_\varepsilon,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\Omega_\varepsilon:=\omega\times (0,\varepsilon),\;\omega\subset \mathbb{R}^{d-1},\;G_\varepsilon [v]=\frac 1{\varepsilon}\int_{\Omega_\varepsilon}g(y',v(y))\,dyNEWLINE\]NEWLINE and \(\mathcal{A}\) is the linear first-order partial differential operator NEWLINE\[NEWLINE\mathcal{A}v:= \sum_{k=1}^d A^{(k)}\partial_k v,\;A^{(1)},\dots,A^{(d)}\in \mathbb{R}^{\ell\times m}.NEWLINE\]NEWLINE The effective behavior of the integral functionals as the thickness of the domain \(\Omega_\varepsilon\) tends to zero is studied together with investigating both upper and lower bounds for the \(\Gamma\)-limit. Under certain conditions, the limit is an integral functional and its explicit formula is given.