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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 4, Pages 88–94 (Mi faa3164)

Brief communications

Homogenization of Elliptic Problems Depending on a Spectral Parameter

T. A. Suslina

St. Petersburg State University, Faculty of Physics

Abstract: In $L_2({\mathbb R}^d;{\mathbb C}^n)$ we consider a strongly elliptic operator $A_\varepsilon$ given by the differential expression $b({\mathbf D})^*g({\mathbf x}/\varepsilon)b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf x})$ is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and $b({\mathbf D})$ is a first-order differential operator. Let ${\mathcal O}\subset {\mathbb R}^d$ be a bounded domain with boundary of class $C^{1,1}$. We also study the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ in $L_2({\mathcal O};{\mathbb C}^n)$ given by the same expression with Dirichlet or Neumann boundary conditions, respectively. We find approximations for the resolvents $(A_\varepsilon -\zeta I)^{-1}$, $(A_{D,\varepsilon} -\zeta I)^{-1}$, and $(A_{N,\varepsilon} -\zeta I)^{-1}$ in the operator ($L_2 \to L_2$)- and ($L_2 \to H^1$)-norms with error estimates depending on the parameters $\varepsilon$ and $\zeta$.

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

 Funding Agency Grant Number Saint Petersburg State University 11.38.63.2012 Russian Foundation for Basic Research 14-01-00760

DOI: https://doi.org/10.4213/faa3164

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English version:
Functional Analysis and Its Applications, 2014, 48:4, 309–313

Bibliographic databases:

UDC: 517.956.2

Citation: T. A. Suslina, “Homogenization of Elliptic Problems Depending on a Spectral Parameter”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 88–94; Funct. Anal. Appl., 48:4 (2014), 309–313

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3164
• https://doi.org/10.4213/faa3164
• http://mi.mathnet.ru/eng/faa/v48/i4/p88

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. M. Meshkova, T. A. Suslina, “Homogenization of Solutions of Initial Boundary Value Problems for Parabolic Systems”, Funct. Anal. Appl., 49:1 (2015), 72–76
2. T. A. Suslina, “Homogenization of elliptic operators with periodic coefficients depending on the spectral parameter”, St. Petersburg Math. J., 27:4 (2016), 651–708
3. Meshkova Yu.M., Suslina T.A., “Two-parametric error estimates in homogenization of second-order elliptic systems in $\mathbb{R}^d$”, Appl. Anal., 95:7, SI (2016), 1413–1448
4. 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
5. 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
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