Interior Hölder and gradient estimates for the homogenization of the linear elliptic equations

Hölder and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius’s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O(?) in the Hölder norm; it is O(?) in W ^1,∞ based on the Avellaneda-Lin’s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.