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We consider the corrector equation associated, in homogenization theory , toa linear second-order elliptic equation in divergence form--$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is alocally perturbed periodic coefficient. The question under study is theexistence (and uniqueness) of the corrector, strictly sublinear at infinity,with gradient in L r if the local perturbation is itself L r , r < +$\infty$.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 thiswell-posedness result. Equations in non-divergence form such as--aij$\partial$iju = f are also considered, along with various extensions. Thecase of general advection-diffusion equations --aij$\partial$iju +bj$\partial$ju = f is postponed until our future work [10]. An appendixcontains a corrigendum to our earlier publication [9].

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