Nonlocal H-convergence for topologically nontrivial domains

Abstract

The notion of nonlocal H-convergence is extended to domains with nontrivial topology, that is, domains with non-vanishing harmonic Dirichlet and/or Neumann fields. If the space of harmonic Dirichlet (or Neumann) fields is infinite-dimensional, there is an abundance of choice of pairwise incomparable topologies generalising the one for topologically trivial Ω. It will be demonstrated that if the domain satisfies the Maxwell compactness property the corresponding natural version of the corresponding (generalised) nonlocal H-convergence topology has no such ambiguity. Moreover, on multiplication operators the nonlocal H-topology coincides with the one induced by (local) H-convergence introduced by Murat and Tartar. The topology is used to obtain nonlocal homogenisation results including convergence of the associated energy for electrostatics. The derived techniques prove useful to deduce a new compactness criterion relevant for nonlinear static Maxwell problems.