  In this paper, we are interested in the periodic homogenization of
quasilinear elliptic equations. We obtain error estimates
$O(\varepsilon^{1/2})$ for a $C^{1,1}$ domain, and $O(\varepsilon^\sigma)$ for
a Lipschitz domain, in which $\sigma\in(0,1/2)$ is close to zero. Based upon
the convergence rates, an interior Lipschitz estimate, as well as a boundary
H\"older estimate can be developed at large scales without any smoothness
assumption, and these will implies reverse H\"older estimates established for a
$C^1$ domain. By a real method developed by Z.Shen \cite{S3}, we consequently
derive a global $W^{1,p}$ estimate for $2\leq p<\infty$. This work may be
regarded as an extension of \cite{MAFHL,S5} to a nonlinear operator, and our
results may be extended to the related Neumann boundary problems without any
real difficulty.
