Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

Abstract

An elliptic fourth-order differential operator $A_\varepsilon$ on $L_2(\mathbb{R}^d;\mathbb{C}^n)$ is studied. Here $\varepsilon >0$ is a small parameter. It is assumed that the operator is given in the factorized form $A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, where $g(\mathbf{x})$ is a Hermitian matrix-valued function periodic with respect to some lattice and $b(\mathbf{D})$ is a matrix second-order differential operator. We make assumptions ensuring that the operator $A_\varepsilon$ is strongly elliptic. The following approximation for the resolvent $(A_\varepsilon + I)^{-1}$ in the operator norm of $L_2(\mathbb{R}^d;\mathbb{C}^n)$ is obtained:

$$(A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2} K_{2}(\varepsilon)+O(\varepsilon^{3}).$$

Here $A^0$ is the effective operator with constant coefficients and $K_{1}$ and $K_{2}(\varepsilon)$ are certain correctors.