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 $ epsiloninmathbb{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 abla u|^{2} dx $, subject to $ ugeq varphi $ on $ S_{varepsilon} $, up to a subsequence, converges weakly in $ H^{1}(D) $ to $ widetilde{u} $ which minimizes the energy functional $int_{D}|gamma abla u|^{2}+int_{Sigma} (u-varphi)^{2}_{-}mu(x) dS_{x}$, where $mu(x)$ depends on the structure of $S_{epsilon}$ and $ varphi $ is any given function in $C^{infty}(overline{D})$.