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

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.

References in corpus (13)

• Nonlocal diffusion and applications
• A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• Nonlocal $H$-convergence
• $\Gamma$-convergence for functionals depending on vector fields. I. Integral representation and compactness
• Leibniz rules and Gauss-Green formulas in distributional fractional spaces
• $\Gamma$-convergence for functionals depending on vector fields. II. Convergence of minimizers
• Nonlocal-interaction vortices
• Homogenization for non-local elliptic operators in both perforated and non-perforated domains
• $H$-convergence for equations depending on monotone operators in Carnot groups
• $G$-convergence of elliptic and parabolic operators depending on vector fields
• Another look at elliptic homogenization