Homogenization of random fractional obstacle problems via $\Gamma$-convergence

TL;DR

This paper proves homogenization results for fractional obstacle problems with randomly sized obstacles in perforated domains using $ ext{Gamma}$-convergence, ergodic theory, and weighted Sobolev norms, offering an alternative proof to existing methods.

Contribution

It introduces a novel approach combining $ ext{Gamma}$-convergence and weighted Sobolev norms to analyze fractional obstacle problems with random obstacles, providing an alternative proof to prior work.

Findings

Homogenization results established for fractional obstacle problems with random obstacles.

Use of a trace-like representation of fractional Sobolev norms in the analysis.

Application of ergodic theorems and joining lemmas in the proof.

Abstract

$\Gamma$-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.