Boundary homogenization of a class of obstacle problems

Abstract

We study homogenization of a boundary obstacle problem on $ C^{1,α} $ domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $γ$. For any $ ε\in\mathbb{R}_+$, $\partial D=Γ\cup Σ$, $Γ\cap Σ=\emptyset $ and $ S_ε\subset Σ$ with suitable assumptions,\ we prove that as $ε$ tends to zero, the energy minimizer $ u^ε $ of $ \int_{D} |γ\nabla u|^{2} dx $, subject to $ u\geq φ$ on $ S_{\varepsilon} $, up to a subsequence, converges weakly in $ H^{1}(D) $ to $ \widetilde{u} $ which minimizes the energy functional $\int_{D}|γ\nabla u|^{2}+\int_Σ (u-φ)^{2}_{-}μ(x) dS_{x}$, where $μ(x)$ depends on the structure of $S_ε$ and $ φ$ is any given function in $C^{\infty}(\overline{D})$.