### 3D-2D asymptotic analysis for micromagnetic thin films

$\Gamma $-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon $ approaches zero of a ferromagnetic thin structure ${\Omega}_{\epsilon}=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {\mathbb{R}}^{2}$, whose energy is given by$${\mathcal{E}}_{\epsilon}\left(\overline{m}\right)=\frac{1}{\epsilon}{\int}_{{\Omega}_{\epsilon}}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}\xb7\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi}_{{\Omega}_{\epsilon}})=0\phantom{\rule{1.0em}{0ex}}\text{on}{\mathbb{R}}^{3},$$and to the constraint$$|\overline{m}|=1\text{on}{\Omega}_{\epsilon},$$where $W$ is any continuous function satisfying $p$-growth assumptions with $p\>1$. Partial results are also obtained in the case $p=1$, under an additional assumption on $W$.