Bounded Correctors in Almost Periodic Homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.