Homogenization of stiff inclusions through network approximation

David Gérard-Varet; Alexandre Girodroux-Lavigne; David Gérard-Varet; Alexandre Girodroux-Lavigne

doi:10.3934/nhm.2022002

Networks and Heterogeneous Media

2022, Volume 17, Issue 2: 163-202. doi: 10.3934/nhm.2022002

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Homogenization of stiff inclusions through network approximation

Université de Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, 75205 Paris CEDEX 13 France

Received: 01 June 2021 Revised: 01 January 2022 Published: 18 February 2022
35R60, 94C05, 78M40

We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.
- Stochastic homogenization,
- network approximation,
- high-contrast media,
- effective conductivity,
- random inclusions
Citation: David Gérard-Varet, Alexandre Girodroux-Lavigne. Homogenization of stiff inclusions through network approximation[J]. Networks and Heterogeneous Media, 2022, 17(2): 163-202. doi: 10.3934/nhm.2022002

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Abstract

We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.

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