$\Gamma$-convergence of oscillating thin obstacles

Yu. O. Koroleva ; M. H. Strömqvist

Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 88-100

Résumé

We consider the minimization problems of obstacle type $$ \min\left\{\int_\Omega|Du|^2\,dx\colon u\ge\psi_\varepsilon\ \text{on}\ P,\ u=0\ \text{on}\ \partial\Omega\right\}, $$ as $\varepsilon\to0$. 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 $P\subset\subset\Omega$ is a subset of the hyper-plane $\{x\in\mathbb R^n\colon x\cdot\eta=0\}$. We assume that $n\ge3$ and that the normal $\eta$ satisfies a generic condition that guarantees certain ergodic properties of the quantity $$ \#\left\{k\in\mathbb Z^n\colon P\cap\{x\colon|x-\varepsilon k|\varepsilon^{n/(n-1)}\}\right\}. $$ Under these hypotheses we compute explicitly the limit functional of the obstacle problem above, which is of the type $$ H^1_0(\Omega)\owns u\mapsto\int_\Omega|Du|^2\,dx+\int_PG(u)\,d\sigma. $$