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Funktsional. Anal. i Prilozhen., 2012, Volume 46, Issue 2, Pages 92–96 (Mi faa3071)  

This article is cited in 10 scientific papers (total in 10 papers)

Brief communications

Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm

M. A. Pakhnin, T. A. Suslina

St. Petersburg State University, Faculty of Physics

Abstract: Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain with boundary of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, consider a matrix elliptic second-order differential operator $A_{D,\varepsilon}$ with Dirichlet boundary condition. Here $\varepsilon >\nobreak0$ is a small parameter; the coefficients of $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The operator $A_{D,\varepsilon}^{-1}$ in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ is approximated with an error of order $\varepsilon^{1/2}$. The approximation is given by the sum of the operator $(A^0_D)^{-1}$ and a first-order corrector. Here $A^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.

Keywords: homogenization of periodic differential operators, effective operator, corrector, operator error estimates

DOI: https://doi.org/10.4213/faa3071

Full text: PDF file (176 kB)
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English version:
Functional Analysis and Its Applications, 2012, 46:2, 155–159

Bibliographic databases:

UDC: 517.956.2
Received: 18.01.2012

Citation: M. A. Pakhnin, T. A. Suslina, “Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm”, Funktsional. Anal. i Prilozhen., 46:2 (2012), 92–96; Funct. Anal. Appl., 46:2 (2012), 155–159

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. A. Pakhnin, T. A. Suslina, “Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm”, Funct. Anal. Appl., 46:2 (2012), 155–159  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. T. A. Suslina, “Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem”, Funct. Anal. Appl., 46:3 (2012), 234–238  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. M. A. Pakhnin, T. A. Suslina, “Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain”, St. Petersburg Math. J., 24:6 (2013), 949–976  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. Suslina T.A., “Homogenization of the Dirichlet problem for elliptic systems: $L^2$-operator error estimates”, Mathematika, 59:2 (2013), 463–476  crossref  mathscinet  zmath  isi  elib  scopus
    5. Suslina T., “Homogenization of the Neumann problem for elliptic systems with periodic coefficients”, SIAM J. Math. Anal., 45:6 (2013), 3453–3493  crossref  mathscinet  zmath  isi  scopus
    6. T. A. Suslina, “Homogenization of Elliptic Problems Depending on a Spectral Parameter”, Funct. Anal. Appl., 48:4 (2014), 309–313  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. T. A. Suslina, “Homogenization of elliptic operators with periodic coefficients depending on the spectral parameter”, St. Petersburg Math. J., 27:4 (2016), 651–708  mathnet  crossref  mathscinet  isi  elib
    8. Meshkova Yu.M. Suslina T.A., “Homogenization of initial boundary value problems for parabolic systems with periodic coefficients”, Appl. Anal., 95:8 (2016), 1736–1775  crossref  mathscinet  zmath  isi  elib  scopus
    9. T. A. Suslina, “Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients”, St. Petersburg Math. J., 29:2 (2018), 325–362  mathnet  crossref  isi  elib
    10. Gu Sh., “Convergence Rates of Neumann Problems For Stokes Systems”, J. Math. Anal. Appl., 457:1 (2018), 305–321  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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