Homogenization of a Boundary Obstacle Problem

We prove the existence of a homogenization limit for solutions of appropriately formulated sequences of boundary obstacle problems for the Laplacian on $C^{1,\alpha }$ domains. Specifically, we prove that the energy minimizers $u_{ϵ}$ of $\int \mathrm{\mid }\mathrm{\nabla }{u}_{ϵ}{\mathrm{\mid }}^{2}dx$, subject to $u \geq \phi$onasubset$S_\epsilon$,convergesweaklyin$H^1$toalimit$\bar{u}$whichminimizestheenergy$\int |\nabla \bar{u}|^2 dx + \int_\Sigma (u-\phi)-^2 \mu(x) dS_x$,$\Sigma \subset \partial D$,iftheobstacleset$S\epsilon$shrinksinanappropriatewaywiththescalingparameter$\epsilon$. This is an extension of a result by Caffarelli and Mellet, which in turn was an extension of a result of Cioranescu and Murat.