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Homogenization of quadratic convolution energies in periodically perforated domains

Andrea Braides ORCID logo EMAIL logo and Andrey Piatnitski ORCID logo


We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

Communicated by Irene Fonseca

Award Identifier / Grant number: E83C18000100006

Funding statement: The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.


[1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal. 18 (1992), no. 5, 481–496. 10.1016/0362-546X(92)90015-7Search in Google Scholar

[2] A. Braides, Γ-convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University, Oxford, 2002. 10.1093/acprof:oso/9780198507840.001.0001Search in Google Scholar

[3] A. Braides, A handbook of Γ-convergence, Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. III, Elsevier/North Holland, Amsterdam (2006), 101–213. 10.1016/S1874-5733(06)80006-9Search in Google Scholar

[4] A. Braides, V. Chiadò Piat and A. Piatnitski, Homogenization of discrete high-contrast energies, SIAM J. Math. Anal. 47 (2015), no. 4, 3064–3091. 10.1137/140975668Search in Google Scholar

[5] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford Lecture Ser. Math. Appl. 12, The Clarendon, New York, 1998. Search in Google Scholar

[6] A. Braides, M. Maslennikov and L. Sigalotti, Homogenization by blow-up, Appl. Anal. 87 (2008), no. 12, 1341–1356. 10.1080/00036810802555458Search in Google Scholar

[7] A. Braides and A. Piatnitski, Homogenization of random convolution energies in heterogeneous and perforated domains, preprint (2019), Search in Google Scholar

[8] D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (1979), no. 2, 590–607. 10.1016/0022-247X(79)90211-7Search in Google Scholar

[9] G. Dal Maso, An Introduction to Γ-convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0327-8Search in Google Scholar

[10] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal. 262 (2012), no. 3, 1274–1308. 10.1016/j.jfa.2011.11.005Search in Google Scholar

[11] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 , SIAM J. Math. Anal. 23 (1992), no. 5, 1081–1098. 10.1137/0523060Search in Google Scholar

[12] E. J. Hruslov, Asymptotic behavior of the solutions of the second boundary value problem in the case of the refinement of the boundary of the domain, Mat. Sb. 106(148) (1978), no. 4, 604–621. Search in Google Scholar

[13] Y. Kondratiev, O. Kutoviy and S. Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 2, 231–258. 10.1142/S0219025708003038Search in Google Scholar

[14] V. A. Marchenko and E. Y. Khruslov, Boundary Problems in Domains with Finely Granulated Boundaries, Naukova Dumka, Kiev, 1974. Search in Google Scholar

[15] A. Piatnitski and E. Zhizhina, Periodic homogenization of nonlocal operators with a convolution-type kernel, SIAM J. Math. Anal. 49 (2017), no. 1, 64–81. 10.1137/16M1072292Search in Google Scholar

[16] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 3, 239–271. 10.1007/BF02838079Search in Google Scholar

Received: 2019-09-28
Revised: 2020-06-04
Accepted: 2020-06-22
Published Online: 2020-07-16
Published in Print: 2022-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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