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


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Vol. 140 of Pure and Applied Mathematics, 2nd edn. Elsevier, Amsterdam (2003)

    Google Scholar 

  2. 2.

    Allaire, G.: Shape Optimization by the Homogenization Method, vol. 146. Springer, Berlin (2012)

    Google Scholar 

  3. 3.

    Andrés, F., Muñoz, J.: Nonlocal optimal design: a new perspective about the approximation of solutions in optimal design. J. Math. Anal. Appl. 429, 288–310 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bellido, J.C., Cueto, J., Mora-Corral, C.: Fractional Piola identity and polyconvexity in fractional spaces, Preprint (2019)

  5. 5.

    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Elbau, P.: Sequential Lower Semi-continuity of Non-local Functionals (2011). arXiv:1104.2686

  8. 8.

    Fernández Bonder, J., Ritorto, A., Salort, A.M.: \(H\)-convergence result for nonlocal elliptic-type problems via Tartar’s method. SIAM J. Math. Anal. 49, 2387–2408 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Focardi, M.: \(\Gamma \)-convergence: a tool to investigate physical phenomena across scales. Math. Methods Appl. Sci. 35(14), 1613–1658 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mengesha, T., Du, Q.: On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28, 3999–4035 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mengesha, T., Du, Q.: Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140, 82–111 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Murat, F., Tartar, L.: \(H\)-convergence, in Topics in the Mathematical Modelling of Composite Materials, Vol. 31 of Progress Nonlinear Differential Equations Applications, pp. 21–43. Birkhäuser, Boston (1997)

    Google Scholar 

  13. 13.

    Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York (1988)

    Google Scholar 

  14. 14.

    Spagnolo, S.: Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 21, 657–699 (1967)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Waurick, M.: A functional analytic perspective to the div-curl lemma. J. Oper. Theory 80, 95–111 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Waurick, M.: Nonlocal \(H\)-convergence. Calc. Var. Partial Differ. Equ. 57, 159 (2018)

    MathSciNet  Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to José C. Bellido.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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

Mathematics Subject Classification

  • 35B27