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

We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form –$\mathrm{\partial }$i(aij$\mathrm{\partial }$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 < +$\mathrm{\infty }$. 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$\mathrm{\partial }$iju = f are also considered, along with various extensions. The case of general advection-diffusion equations --aij$\mathrm{\partial }$iju + bj$\mathrm{\partial }$ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].