Regularity theory of elliptic systems in $varepsilon$-scale flat domains

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called $varepsilon$-scale flatness condition, which could be arbitrarily rough below $varepsilon$-scale. This particularly generalizes Kenig and Pranges work in [32] and [33] by a quantitative approach.