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This paper concerns with a family of elliptic systems of linear elasticitywith rapidly oscillating periodic coefficients, arising in the theory ofhomogenization. We establish uniform optimal regularity estimates for solutionsof Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. Theproof relies on a boundary Korn inequality for solutions of systems of linearelasticity and uses a large-scale Rellich estimate obtained in\cite{Shen-2016}.

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity