Boundary homogenization of a class of obstacle problems

Abstract

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 $\epsilon \in {\mathbb{R}}_{+}$, $\partial D=\Gamma \cup \Sigma$, $\Gamma \cap \Sigma =\varnothing$ 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 $u\ge \phi$ on $S_{\epsilon }$, up to a subsequence, converges weakly in $H^1(D)$ to $\tilde{u}$ which minimizes the energy functional ${\int }_D|\gamma \nabla u{|}^2+{\int }_{\Sigma }(u-\phi {)}_{-}^2\mu (x)d{S}_x$, where $\mu (x)$ depends on the structure of $S_{\epsilon }$ and $\phi$ is any given function in $C^{\infty }(\overline{D})$.