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Large-Scale Lipschitz Estimates for Elliptic Systems with Periodic High-Contrast Coefficients

This paper is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the contrast ratio $δ^2$ for $0<δ<\infty$. Our study also covers the case of soft inclusions ($δ=0$) as well as the case of stiff inclusions ($δ=\infty$). The large-scale Lipschitz estimate, together with classical local estimates, allows us to establish explicit bounds for the matrix of fundamental solutions and its derivatives.