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The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for d ≥ 4 , we prove that the L p Neumann and L p Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2 ( d − 1 ) d + 1 − δ < p < 2 + δ and 2 − δ < p < 2 ( d − 1 ) d − 3 + δ respectively. [ABSTRACT FROM AUTHOR]