A simple characterization of H-convergence for a class of nonlocal problems

Abstract

This is a follow-up of the paper J. Fernández-Bonder, A. Ritorto and A. Salort, H-convergence result for nonlocal elliptic-type problems via Tartar’s method, SIAM J. Math. Anal., 49 (2017), pp. 2387–2408, where the classical concept of H-convergence was extended to fractional \(p\)-Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak-\(*\) convergence of the coefficients is an equivalent condition for H-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local \(p\)-Laplacian case.

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Acknowledgements

We thank an anonymous referee for noticing us about reference [9]. AE’s research is financially supported by the Villum Fonden through the Villum Investigator Project InnoTop. The work of JCB is funded by FEDER EU and Ministerio de Economía y Competitividad (Spain) through Grant MTM2017-83740-P.

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Correspondence to José C. Bellido.

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Bellido, J.C., Evgrafov, A. A simple characterization of H-convergence for a class of nonlocal problems. Rev Mat Complut 34, 175–183 (2021). https://doi.org/10.1007/s13163-020-00349-9

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Keywords

  • Homogenization of nonlocal problems
  • H-convergence
  • G-convergence

Mathematics Subject Classification

  • 35B27