On correctors for linear elliptic homogenization in the presence of local defects
Xavier Blanc, Claude Le Bris, Pierre Louis Lions · Communications in Partial Differential Equations · 2018
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 Lr if the local perturbation is itself Lr, r<+∞. This work follows up on previous works of ours, 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 to a future work. An appendix contains a corrigendum to one of our earlier publication.