Homogenization of a Boundary Obstacle Problem

TL;DR

This paper establishes the homogenization limit for boundary obstacle problems involving the Laplacian on smooth domains, showing convergence of energy minimizers to a modified energy problem with a boundary measure.

Contribution

It extends previous homogenization results to boundary obstacle problems, providing a rigorous limit description for solutions with shrinking obstacle sets.

Findings

Energy minimizers converge weakly in H^1 to a limit involving a boundary measure.

The limit problem includes an additional boundary integral term with a measure li(x).

The results generalize prior work by Caffarelli, Mellet, Cioranescu, and Murat.

Abstract

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