Homogenization of a Boundary Obstacle Problem

Abstract

We prove the existence of a homogenization limit for solutions of appropriately formulated sequences of boundary obstacle problems for the Laplacian on $C^{1,α}$ domains. Specifically, we prove that the energy minimizers $u_ε$ of $\int |\nabla u_ε|^2 dx$, subject to $u \geq ϕ$ on a subset $S_ε$, converges weakly in $H^1$ to a limit $\bar{u}$ which minimizes the energy $\int |\nabla \bar{u}|^2 dx + \int_Σ(u-ϕ)_-^2 μ(x) dS_x$, $Σ\subset \partial D$, if the obstacle set $S_ε$ shrinks in an appropriate way with the scaling parameter $ε$. This is an extension of a result by Caffarelli and Mellet, which in turn was an extension of a result of Cioranescu and Murat.