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A periodic homogenization problem with defects rare at infinity

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  • 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.


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  • 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 $

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