  We study homogenization of a boundary obstacle problem on $ C^{1,\alpha} $
domain $D$ for some elliptic equations with uniformly elliptic coefficient
matrices $\gamma$. For any $ \epsilon\in\mathbb{R}_+$, $\partial D=\Gamma \cup
\Sigma$, $\Gamma \cap \Sigma=\emptyset $ and $ S_{\epsilon}\subset \Sigma $
with suitable assumptions,\ we prove that as $\epsilon$ tends to zero, the
energy minimizer $ u^{\epsilon} $ of $ \int_{D} |\gamma\nabla u|^{2} dx $,
subject to $ u\geq \varphi $ on $ S_{\varepsilon} $, up to a subsequence,
converges weakly in $ H^{1}(D) $ to $ \widetilde{u} $ which minimizes the
energy functional $\int_{D}|\gamma\nabla u|^{2}+\int_{\Sigma}
(u-\varphi)^{2}_{-}\mu(x) dS_{x}$, where $\mu(x)$ depends on the structure of
$S_{\epsilon}$ and $ \varphi $ is any given function in
$C^{\infty}(\overline{D})$.
