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This article is cited in 13 scientific papers (total in 13 papers)
Brief communications
Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
V. A. Sloushch, T. A. Suslina St. Petersburg State University, St. Petersburg, Russia
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.
Keywords:
periodic differential operators, homogenization, operator error estimates, effective operator, corrector.
Received: 07.07.2020 Revised: 09.07.2020 Accepted: 12.07.2020
Citation:
V. A. Sloushch, T. A. Suslina, “Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account”, Funktsional. Anal. i Prilozhen., 54:3 (2020), 94–99; Funct. Anal. Appl., 54:3 (2020), 224–228
Linking options:
https://www.mathnet.ru/eng/faa3807https://doi.org/10.4213/faa3807 https://www.mathnet.ru/eng/faa/v54/i3/p94
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