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