Characterisation of homogenisation for nonlocal diffusion by local topologies

TL;DR

This paper studies the homogenisation of nonlocal fractional divergence form problems with oscillatory coefficients, characterising convergence through local and nonlocal topologies, and applies these results to fractional heat equations.

Contribution

It introduces a comprehensive framework for understanding homogenisation in nonlocal fractional problems using classical and nonlocal $H$-convergence and Schur topologies.

Findings

Characterisation of coefficient convergence via $H$-convergence and weak-* convergence.

Application to homogenisation of fractional heat equations.

Extension of nonlocal $H$-convergence concepts to fractional divergence problems.

Abstract

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the fractional divergence form problem and weak-$\ast$ convergence on the complement caused by the nonlocality of the differential operators. The results are further described in the light of nonlocal $H$-convergence as introduced in [Waurick, Calc Var PDEs, 57, 2018] and certain Schur topologies. Applications to symmetric coefficients and a homogenisation problem for a fractional heat type equation are provided.