Boundary Homogenization of a Class of Obstacle Problems

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