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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2204.12315 |
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| _version_ | 1866913569147191296 |
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| author | Waurick, Marcus |
| author_facet | Waurick, Marcus |
| contents | 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'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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_12315 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonlocal $H$-convergence for topologically nontrivial domains Waurick, Marcus Analysis of PDEs Mathematical Physics Functional Analysis Primary 35B27, Secondary 35Q61, 74Q05 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'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. |
| title | Nonlocal $H$-convergence for topologically nontrivial domains |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Primary 35B27, Secondary 35Q61, 74Q05 |
| url | https://arxiv.org/abs/2204.12315 |