Regularity theory of elliptic systems in $\epsilon$-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 $\epsilon$-scale flatness condition, which could be arbitrarily rough below $\epsilon$-scale. This particularly generalizes Kenig and Prange’s work in [32] and [33] by a quantitative approach.