  The main purpose of this work is to study uniform regularity estimates for a
family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$,
arising in the theory of homogenization, with rapidly oscillating periodic
coefficients. We establish sharp $W^{1,p}$ estimates, Lipschitz estimates, and
nontangential maximal function estimates, which are uniform in the parameter
$\varepsilon$, on solutions with Neumann boundary conditions in $C^{1,\alpha}$
domains.
