  In this paper, we consider a family of second-order elliptic systems subject
to a periodically oscillating Robin boundary condition. We establish the
qualitative homogenization theorem on any Lipschitz domains satisfying a
non-resonance condition. We also use the quantitative estimates of oscillatory
integrals to obtain the dimension-dependent convergence rates in $L^2$,
assuming that the domain is smooth and strictly convex.
