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In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form  ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution  u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution  u 0 satisfies than the classical homogenization theory needs.

A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients