Uniform estimates for systems of elasticity in homogenization

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 omega must satisfy in order for the weighted W-1,W-2(omega) estimates for weak solutions of Neumann problems to be true. Moreover, in any Lipschitz domain, under the assumption that the coefficient A is Holder continuous, we prove that the uniform W- 1,W-p estimates for solutions to the Neumann problem hold for 2d/d-1 + delta. As a by-product, in non-periodic setting with A is an element of V MO, we are able to show that the W-1,W-p estimates hold for 2d/d+1 - delta < p < 2d/d-1 + delta. 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 < p0 < 2(d-1/d-2) to the range p0 < p < 2(d-1)/d-2 to the range p02 regularity estimate. (c) 2024 Elsevier Inc. All rights reserved.