$H$-compactness for nonlocal linear operators in fractional divergence form

TL;DR

This paper investigates the $H$-convergence of nonlocal linear operators in fractional divergence form, introducing a novel compactness approach that overcomes limitations of classical methods, and establishing equivalence with $ ext{Gamma}$-convergence under symmetry.

Contribution

It develops a new compactness argument for $H$-convergence of nonlocal operators, extending classical results to fractional divergence form and linking it to energy $ ext{Gamma}$-convergence when symmetry is assumed.

Findings

Established $H$-compactness for nonlocal fractional divergence operators.

Extended the equivalence between $H$-convergence and $ ext{Gamma}$-convergence under symmetry.

Provided a novel approach bypassing classical localization limitations.

Abstract

We study the $H$-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. If symmetry is also assumed, we extend the equivalence between the $H$-convergence of the operators and the $\Gamma$-convergence of the associated energies.