A periodic homogenization problem with defects rare at infinity

Rémi Goudey; Rémi Goudey

doi:10.3934/nhm.2022014

Networks and Heterogeneous Media

2022, Volume 17, Issue 4: 547-592. doi: 10.3934/nhm.2022014

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A periodic homogenization problem with defects rare at infinity

Rémi Goudey

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

Received: 01 September 2021 Revised: 01 February 2022 Published: 28 March 2022
Primary:35B27,35J15;Secondary:74Q15

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.
- Homogenization,
- elliptic PDEs,
- defects,
- corrector equation,
- convergence estimates
Citation: Rémi Goudey. A periodic homogenization problem with defects rare at infinity[J]. Networks and Heterogeneous Media, 2022, 17(4): 547-592. doi: 10.3934/nhm.2022014

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Abstract

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.

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