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For a fixed bounded Lipschitz domain and a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we investigate a necessary and sufficient condition that an A 1 weight ω must satisfy in order for the weighted W 1 , 2 (ω) estimates for weak solutions of Neumann problems to be true. Moreover, in any Lipschitz domain, under the assumption that the coefficient A is Hölder continuous, we prove that the uniform W 1 , p estimates for solutions to the Neumann problem hold for 2 d d + 1 − δ < p < 2 d d − 1 + δ. As a by-product, in non-periodic setting with A ∈ V M O , we are able to show that the W 1 , p estimates hold for 2 d d + 1 − δ < p < 2 d d − 1 + δ. The ranges are sharp for d = 2 , 3. Finally, we prove an extrapolation result for L p Dirichlet problems for systems of linear elasticity. Specifically, we extrapolate from solvability for 1 < p 0 < 2 (d − 1) d − 2 to the range p 0 < p < 2 (d − 1) d − 2 + δ. The novelty is that the method avoids using the L 2 regularity estimate. [ABSTRACT FROM AUTHOR]

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]

For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in L 2 . Both initial-Dirichlet and initial-Neumann problems are studied. [ABSTRACT FROM AUTHOR]

Abstract: Let be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform estimate in a Lipschitz domain Ω in for solutions to the Dirichlet problem: in Ω and on ∂Ω, where and C, are constants independent of . The ranges are sharp for or 3. [Copyright &y& Elsevier]