The main purpose of this work is to study uniform regularity estimates for afamily of elliptic operators ${mathcal{L}_arepsilon, arepsilon>0}$,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$arepsilon$, on solutions with Neumann boundary conditions in $C^{1,lpha}$domains.
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