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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Vol. 140 of Pure and Applied Mathematics, 2nd edn. Elsevier, Amsterdam (2003)
Allaire, G.: Shape Optimization by the Homogenization Method, vol. 146. Springer, Berlin (2012)
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)
Bellido, J.C., Cueto, J., Mora-Corral, C.: Fractional Piola identity and polyconvexity in fractional spaces, Preprint (2019)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
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)
Elbau, P.: Sequential Lower Semi-continuity of Non-local Functionals (2011). arXiv:1104.2686
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)
Focardi, M.: \(\Gamma \)-convergence: a tool to investigate physical phenomena across scales. Math. Methods Appl. Sci. 35(14), 1613–1658 (2012)
Mengesha, T., Du, Q.: On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28, 3999–4035 (2015)
Mengesha, T., Du, Q.: Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140, 82–111 (2016)
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)
Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York (1988)
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)
Waurick, M.: A functional analytic perspective to the div-curl lemma. J. Oper. Theory 80, 95–111 (2018)
Waurick, M.: Nonlocal \(H\)-convergence. Calc. Var. Partial Differ. Equ. 57, 159 (2018)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-020-00349-9