Advanced Search
Article Contents
Article Contents

A periodic homogenization problem with defects rare at infinity

Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary:35B27,35J15;Secondary:74Q15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Prototype perturbation in dimension $ d = 1 $

    Figure 2.  Example of points in ambient dimension 2 that satisfy our assumptions along with their associated Voronoi diagram

    Figure 3.  Example for the choice of the open subset $ W_x $ when $ d = 2 $

  • [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. BlancM. 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. BlancC. 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. BlancC. 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. BlancC. 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. BlancF. 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. GiaquintaMultiple 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.
  • 加载中



Article Metrics

HTML views(806) PDF downloads(188) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint