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On correctors for linear elliptic homogenization in the presence of local defects

Abstract : We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form −∂i(aij∂ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +∞. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as −aij∂iju = f are also considered, along with various extensions. The case of general advection-diffusion equations −aij∂iju + bj∂ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].
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https://hal.archives-ouvertes.fr/hal-01697104
Contributor : Xavier Blanc <>
Submitted on : Tuesday, January 30, 2018 - 10:35:59 PM
Last modification on : Thursday, January 7, 2021 - 4:00:05 PM
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Xavier Blanc, Claude Le Bris, Pierre Louis Lions. On correctors for linear elliptic homogenization in the presence of local defects. 2018. ⟨hal-01697104⟩

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