Nonlocal H-convergence

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Abstract

We introduce the concept of nonlocal H-convergence. For this we employ the theory of abstract closed complexes of operators in Hilbert spaces. We show uniqueness of the nonlocal H-limit as well as a corresponding compactness result. Moreover, we provide a characterisation of the introduced concept, which implies that local and nonlocal H-convergence coincide for multiplication operators. We provide applications to both nonlocal and nonperiodic fully time-dependent 3D Maxwell's equations on rough domains. The material law for Maxwell's equations may also rapidly oscillate between eddy current type approximations and their hyperbolic non-approximated counter parts. Applications to models in nonlocal response theory used in quantum theory and the description of meta-materials, to fourth order elliptic problems as well as to homogenisation problems on Riemannian manifolds are provided.
Original languageEnglish
Article number159
Number of pages46
JournalCalculus of Variations and Partial Differential Equations
Volume57
Early online date29 Sep 2018
DOIs
Publication statusPublished - 30 Dec 2018

Keywords

  • homogenisation
  • H-convergence
  • nonlocal coefficients
  • complexes of operators
  • evolutionary equations
  • equations of mixed type
  • Maxwell's equations
  • plate equation
  • partial differential equations on manifolds

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