Boundary homogenization of a class of obstacle problems

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 $C^{\infty}(\overline{D})$.