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doi: 10.3934/nhm.2022014
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## A periodic homogenization problem with defects rare at infinity

 École des Ponts ParisTech and INRIA Paris, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France

Received  September 2021 Revised  February 2022 Early access March 2022

We consider a homogenization problem for the diffusion equation $-\operatorname{div}\left(a_{\varepsilon} \nabla u_{\varepsilon} \right) = f$ when the coefficient $a_{\varepsilon}$ is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of $u_{\varepsilon}$ to its homogenized limit.

Citation: Rémi Goudey. A periodic homogenization problem with defects rare at infinity. Networks and Heterogeneous Media, doi: 10.3934/nhm.2022014
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Communications on Pure and Applied Mathematics, 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607. [3] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization II: Equations in non-divergence form, Communications on Pure and Applied Mathematics, 42 (1989), 139-172.  doi: 10.1002/cpa.3160420203. [4] M. Avellaneda and F.-H. Lin, $L^p$ bounds on singular integrals in homogenization, Communications on Pure and Applied Mathematics, 44 (1991), 897-910.  doi: 10.1002/cpa.3160440805. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978. [6] X. Blanc, M. Josien and C. Le Bris, Precised approximations in elliptic homogenization beyond the periodic setting, Asymptotic Analysis, 116 (2020), 93-137.  doi: 10.3233/ASY-191537. [7] X. Blanc, C. Le Bris and P.-L. Lions, On correctors for linear elliptic homogenization in the presence of local defects: The case of advection-diffusion, Journal de Mathématiques Pures et Appliquées, 124 (2019), 106–122. doi: 10.1016/j.matpur.2018.04.010. [8] X. Blanc, C. Le Bris and P.-L. Lions, On correctors for linear elliptic homogenization in the presence of local defects, Communications in Partial Differential Equations, 43 (2018), 965-997.  doi: 10.1080/03605302.2018.1484764. [9] X. Blanc, C. Le Bris and P.-L. Lions, Local profiles for elliptic problems at different scales: Defects in, and interfaces between periodic structures, Communications in Partial Differential Equations, 40 (2015), 2173-2236.  doi: 10.1080/03605302.2015.1043464. [10] X. Blanc, C. Le Bris and P.-L. Lions, A possible homogenization approach for the numerical simulation of periodic microstructures with defects, Milan Journal of Mathematics, 80 (2012), 351-367.  doi: 10.1007/s00032-012-0186-7. [11] X. Blanc, F. Legoll and A. Anantharaman, Asymptotic behaviour of Green functions of divergence form operators with periodic coefficients, Applied Mathematics Research Express, 2013 (2013), 79-101.  doi: 10.1093/amrx/abs013. [12] J. Deny and J.-L. Lions, Les espaces du type de Beppo Levi, Annales de l'institut Fourier, 5 (1954), 305-370.  doi: 10.5802/aif.55. [13] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983. [15] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni Della Normale, Pisa, 2012. doi: 10.1007/978-88-7642-443-4. [16] R. Goudey, A periodic homogenization problem with defects rare at infinity, preprint, arXiv: 2109.05506. [17] R. Goudey, PhD Thesis, in preparation. [18] M. Gruter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Mathematica, 37 (1982), 303-342.  doi: 10.1007/BF01166225. [19] V. V. Jikov, S. M Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. [20] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Parts 1 & 2, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/s0294-1449(16)30422-x. [21] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, Berlin Heidelberger, 2009. doi: 10.1007/978-3-642-05195-1.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Communications on Pure and Applied Mathematics, 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607. [3] M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization II: Equations in non-divergence form, Communications on Pure and Applied Mathematics, 42 (1989), 139-172.  doi: 10.1002/cpa.3160420203. [4] M. Avellaneda and F.-H. Lin, $L^p$ bounds on singular integrals in homogenization, Communications on Pure and Applied Mathematics, 44 (1991), 897-910.  doi: 10.1002/cpa.3160440805. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978. [6] X. Blanc, M. Josien and C. Le Bris, Precised approximations in elliptic homogenization beyond the periodic setting, Asymptotic Analysis, 116 (2020), 93-137.  doi: 10.3233/ASY-191537. [7] X. Blanc, C. Le Bris and P.-L. Lions, On correctors for linear elliptic homogenization in the presence of local defects: The case of advection-diffusion, Journal de Mathématiques Pures et Appliquées, 124 (2019), 106–122. doi: 10.1016/j.matpur.2018.04.010. [8] X. Blanc, C. Le Bris and P.-L. Lions, On correctors for linear elliptic homogenization in the presence of local defects, Communications in Partial Differential Equations, 43 (2018), 965-997.  doi: 10.1080/03605302.2018.1484764. [9] X. Blanc, C. Le Bris and P.-L. Lions, Local profiles for elliptic problems at different scales: Defects in, and interfaces between periodic structures, Communications in Partial Differential Equations, 40 (2015), 2173-2236.  doi: 10.1080/03605302.2015.1043464. [10] X. Blanc, C. Le Bris and P.-L. Lions, A possible homogenization approach for the numerical simulation of periodic microstructures with defects, Milan Journal of Mathematics, 80 (2012), 351-367.  doi: 10.1007/s00032-012-0186-7. [11] X. Blanc, F. Legoll and A. Anantharaman, Asymptotic behaviour of Green functions of divergence form operators with periodic coefficients, Applied Mathematics Research Express, 2013 (2013), 79-101.  doi: 10.1093/amrx/abs013. [12] J. Deny and J.-L. Lions, Les espaces du type de Beppo Levi, Annales de l'institut Fourier, 5 (1954), 305-370.  doi: 10.5802/aif.55. [13] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983. [15] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Edizioni Della Normale, Pisa, 2012. doi: 10.1007/978-88-7642-443-4. [16] R. Goudey, A periodic homogenization problem with defects rare at infinity, preprint, arXiv: 2109.05506. [17] R. Goudey, PhD Thesis, in preparation. [18] M. Gruter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Mathematica, 37 (1982), 303-342.  doi: 10.1007/BF01166225. [19] V. V. Jikov, S. M Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. [20] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Parts 1 & 2, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/s0294-1449(16)30422-x. [21] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, Berlin Heidelberger, 2009. doi: 10.1007/978-3-642-05195-1.
Prototype perturbation in dimension $d = 1$
Example of points in ambient dimension 2 that satisfy our assumptions along with their associated Voronoi diagram
Example for the choice of the open subset $W_x$ when $d = 2$
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