  We establish interior Lipschitz estimates at the macroscopic scale for
solutions to systems of linear elasticity with rapidly oscillating periodic
coefficients and mixed boundary conditions in domains periodically perforated
at a microscopic scale $\varepsilon$ by establishing $H^1$-convergence rates
for such solutions. The interior estimates are derived directly without the use
of compactness via an argument presented in [3] that was adapted for elliptic
equations in [2] and [11]. As a consequence, we derive a Liouville type
estimate for solutions to the systems of linear elasticity in unbounded
periodically perforated domains.
