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Funktsional'nyi Analiz i ego Prilozheniya, 2020, Volume 54, Issue 3, Pages 94–99
DOI: https://doi.org/10.4213/faa3807
(Mi faa3807)
 

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
References:
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.
Funding agency Grant number
Russian Science Foundation 17-11-01069
This work was supported by the Russian Science Foundation, project no. 17-11-01069.
Received: 07.07.2020
Revised: 09.07.2020
Accepted: 12.07.2020
English version:
Functional Analysis and Its Applications, 2020, Volume 54, Issue 3, Pages 224–228
DOI: https://doi.org/10.1134/S0016266320030077
Bibliographic databases:
Document Type: Article
UDC: 517.956.2
Language: Russian
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
Citation in format AMSBIB
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\paper Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
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\vol 54
\issue 3
\pages 94--99
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\jour Funct. Anal. Appl.
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\pages 224--228
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  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Функциональный анализ и его приложения Functional Analysis and Its Applications
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