Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
Vladimir Anatolevich Sloushch, Tatiana Aleksandrovna Suslina · Functional Analysis and Its Applications · 2020
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.