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

We study the $H$-convergence problem for a class 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 show that the $H$-compactness can be equivalently obtained through the $\Gamma$-convergence of the associated energies.