A periodic homogenization problem with defects rare at infinity

Rémi Goudey

doi:10.3934/nhm.2022014

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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: September 2021

Revised: February 2022

Early access: March 2022

Published: August 2022

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

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Figure 1. Prototype perturbation in dimension $ d = 1 $

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

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

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