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

Abstract

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form $\left(\partial /\partial x_i\right)\left(a^{ij}\left(x/\epsilon ,x\right)\left(\partial u^{\epsilon }\left(x\right)/\partial x_j\right)\right)=f\left(x\right)$ . 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.