Boundary Homogenization of a Class of Obstacle Problems

Abstract

We study the homogenization of a boundary obstacle problem on a $C^{1,α}$-domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $\gamma.$ For any $\epsilon \in \mathbb{R}_+,$ $∂D=\Gamma ∪Σ,$ $\Gamma ∩Σ=∅$ and $S_{\epsilon}\subset Σ$ with suitable assumptions, we prove that as $\epsilon$ tends to zero, the energy minimizer $u^{\epsilon}$ of $\int_D |\gamma ∇u|^2dx,$ subject to $u≥\varphi$ on $S_{\epsilon},$ up to a subsequence, converges weakly in $H^1 (D)$ to $\tilde{u},$ which minimizes the energy functional $$\int_D |\gamma∇u|^2+ \int_Σ (u−\varphi)^2\_\mu (x)dS_x,$$ where $\mu (x)$ depends on the structure of $S_{\epsilon}$ and $\varphi$ is any given function in $C^∞(\overline{D}).$