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