  This paper concerns with a family of elliptic systems of linear elasticity
with rapidly oscillating periodic coefficients, arising in the theory of
homogenization. We establish uniform optimal regularity estimates for solutions
of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The
proof relies on a boundary Korn inequality for solutions of systems of linear
elasticity and uses a large-scale Rellich estimate obtained in
\cite{Shen-2016}.
