Aperiodic fractional obstacle problems

TL;DR

This paper analyzes the asymptotic behavior of fractional obstacle problems in complex aperiodic settings, using Gamma-convergence, with applications to physical phenomena involving non-local energies.

Contribution

It extends the understanding of obstacle problems for fractional energies to aperiodic and random environments, including random obstacles and Delone sets.

Findings

Asymptotic behavior characterized via Gamma-convergence

Results applicable to physical phenomena with non-local energies

Analysis includes obstacles with random sizes, shapes, and distributions

Abstract

We determine the asymptotic behaviour of (bilateral) obstacle problems for fractional energies in rather general aperiodic settings via Gamma-convergence arguments. As further developments we consider obstacles with random sizes and shapes located on points of standard lattices, and the case of random homothetics obstacles centered on random Delone sets of points. Obstacle problems for non-local energies occur in several physical phenomenona, for which our results provide a description of the first order asympotitc behaviour.