Boundary Homogenization of a Class of Obstacle Problems

We study the homogenization of a boundary obstacle problem on a C1,α-domain D for some elliptic equations with uniformly elliptic coefficient matrices γ. For any ϵ∈R+, ∂D=Γ∪Σ, Γ∩Σ=∅ and Sϵ⊂Σ with suitable assumptions, we prove that as ϵ tends to zero, the energy minimizer uϵ of ∫D|γ∇u|2dx, subject to u≥φ on Sϵ, up to a subsequence, converges weakly in H1(D) to ~u, which minimizes the energy functional ∫D|γ∇u|2+∫Σ(u−φ)2−μ(x)dSx, where μ(x) depends on the structure of Sϵ and φ is any given function in C∞¯¯¯¯¯D.