Regularity of the correctors and local gradient estimate of the homogenization for the elliptic equation: linear periodic case

$C^\alpha$ and $W^{1,\infty}$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation with piecewise smooth coefficients. If the data are smooth enough, the error of the first-order expansion for piecewise smooth coefficients is locally $O(\epsilon)$ in the H\"older norm; it is locally $O(\epsilon)$ in $W^{1,\infty}$ when coefficients are Lipschitz continuous. It can be partly extended to the nonlinear parabolic equation.