Homogenization of random fractional obstacle problems via $Γ$-convergence

Abstract

$Γ$-convergence methods are used to prove homogenization results for fractional obstacle problems in periodically perforated domains. The obstacles have random sizes and shapes and their capacity scales according to a stationary ergodic process. We use a trace-like representation of fractional Sobolev norms in terms of weighted Sobolev energies established by Caffarelli and Silvestre, a weighted ergodic theorem and a joining lemma in varying domains following the approach by Ansini and Braides. Our proof is alternative to those contained in the papers by Caffarelli and Mellet.