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arXiv · 2204.12315

Nonlocal $H$-convergence for topologically nontrivial domains

Marcus Waurick
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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 $\Omega$. It will be demonstrated that if the domain satisfies the Maxwell's 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.

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Marcus Waurick. 2022-04-26. Nonlocal $H$-convergence for topologically nontrivial domains. https://arxiv.org/abs/2204.12315

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