Highly Oscillating Thin Obstacles

Abstract

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane $\Gamma$ in $\mathbb{R}^n$ and a periodic perforation $\mathcal{T}_\varepsilon$ of $\mathbb{R}^n$, depending on a small parameter $\varepsilon>0$. As $\varepsilon\to 0$, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type $\{k\alpha\}_{k=1}^\infty$, $\alpha\in\R$, we are able to extract rather precise information about the set $\Gamma\cap\mathcal{T}_\varepsilon$. As $\varepsilon\to0$, we determine the limit $u$ of the solution $u_\varepsilon$ to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behaviour for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in $\R^n$.