# American Institute of Mathematical Sciences

March  2020, 15(1): 125-142. doi: 10.3934/nhm.2020006

## A remark about the periodic homogenization of certain composite fibered media

 1 Laboratoire Jacques-Louis Lions, Boite courrier 187, Sorbonne Université, 4 place Jussieu, 75252 Paris cedex 05, France 2 Institut de Mathématiques de Marseille (I2M), UMR 7373, Aix-Marseille Univ, CNRS, Centrale Marseille, CMI, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France

* Corresponding author: Ali Sili

Received  May 2019 Revised  September 2019 Published  December 2019

We explain in this paper the similarity arising in the homogenization process of some composite fibered media with the problem of the reduction of dimension $3d-1d$. More precisely, we highlight the fact that when the homogenization process leads to a local reduction of dimension, studying the homogenization problem in the reference configuration domain of the composite amounts to the study of the corresponding reduction of dimension in the reference cell. We give two examples in the framework of the thermal conduction problem: the first one concerns the reduction of dimension in a thin parallelepiped of size $\varepsilon$ containing another thinner parallelepiped of size $r_ \varepsilon \ll \varepsilon$ playing a role of a "hole". As in the homogenization, the one-dimensional limit problem involves a "strange term". In addition both limit problems have the same structure. In the second example, the geometry is similar but now we assume a high contrast between the conductivity (of order $1$) in the small parallelepiped of size $r_ \varepsilon : = r \varepsilon$, for some fixed $r$ ($0 < r < \frac{1}{2}$) and the conductivity (of order $\varepsilon^2$) in the big parallelepiped of size $\varepsilon$. We prove that the limit problem is a nonlocal problem and that it has the same structure as the corresponding periodic homogenized problem.

Citation: François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media. Networks & Heterogeneous Media, 2020, 15 (1) : 125-142. doi: 10.3934/nhm.2020006
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##### References:
 [1] Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers. Networks & Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709 [2] Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025 [3] Frédéric Legoll, William Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 1-27. doi: 10.3934/dcdss.2015.8.1 [4] Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092 [5] Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29 [6] Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks & Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127 [7] Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279 [8] Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020 [9] Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. Mathematical Foundations of Computing, 2019, 2 (2) : 95-106. doi: 10.3934/mfc.2019008 [10] Lyudmila Grigoryeva, Juan-Pablo Ortega. Dimension reduction in recurrent networks by canonicalization. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021028 [11] Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228 [12] Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003 [13] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [14] María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004 [15] Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91 [16] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385 [17] Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129 [18] Martin Kružík, Ulisse Stefanelli, Chiara Zanini. Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5999-6013. doi: 10.3934/dcds.2015.35.5999 [19] Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1 [20] Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks & Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025

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