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