  In this paper, we extend the uniform regularity estimates obtained by M.
Avellanda and F. Lin in the paper of Compactness methods in the theory of
homogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the more
general second order elliptic systems in divergences form, with rapidly
oscillating periodic coefficients. We establish not only sharp $W^{1,p}$
estimates, Holder estimates, Lipschitz estimates and non-tangential maximal
function estimates for the Dirichelt problem on a $C^{1,\eta}$ domain, but also
a sharp $O(\varepsilon)$ convergence rate in $H_0^1(\Omega)$ by virtue of the
Dirichlet correctors. The well known compactness methods are not employed here,
instead we construct the transformations to make full use of the corresponding
results developed by M. Avellanda and F. Lin.
