Γ-convergence of Oscillating Thin Obstacles

Abstract

We consider the minimization problems of obstacle type

\[ \min \left\{ \int_{\Omega} |Du|^2 \, dx : u \ge \psi_\varepsilon \text{ on } P, \; u = 0 \text{ on } \partial \Omega \right\}, \]

as \( \varepsilon \to 0 \). Here \( \Omega \) is a bounded domain in \( \mathbb{R}^n \), \( \psi_\varepsilon \) is a periodic function of period \( \varepsilon \), constructed from a fixed function \( \psi \), and \( P \subset \Omega \) is a subset of the hyper-plane \( \{x \in \mathbb{R}^n : x \cdot \eta = 0 \} \). We assume that \( n \ge 3 \) and that the normal \( \eta \) satisfies a generic condition that guarantees certain ergodic properties of the quantity

\[ \# \left\{ k \in \mathbb{Z}^n : P \cap \{ x : |x - \varepsilon k| < \varepsilon^{n/(n-1)} \} \right\}. \]

Under these hypotheses we compute explicitly the limit functional of the obstacle problem above, which is of the type

\[ H_0^1(\Omega) \ni u \mapsto \int_{\Omega} |Du|^2 \, dx + \int_P G(u) \, d\sigma. \]