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Algebra i Analiz, 2010, Volume 22, Issue 5, Pages 69–103 (Mi aa1205)  

This article is cited in 6 scientific papers (total in 6 papers)

Research Papers

Homogenization of periodic differential operators of high order

N. A. Veniaminov

St. Petersburg State University, Faculty of Physics, St. Petersburg, Russia

Abstract: A periodic differential operator of the form $A_\varepsilon=(\mathbf D^p)^*g(\mathbf x/\varepsilon)\mathbf D^p$ is considered on $L_2(\mathbb R^d)$; here $g(x)$ is a positive definite symmetric tensor of order $2p$ periodic with respect to a lattice $\Gamma$. The behavior of the resolvent of the operator $A_\varepsilon$ as $\varepsilon\to0$ is studied. It is shown that the resolvent $(A_\varepsilon+I)^{-1}$ converges in the operator norm to the resolvent of the effective operator $A^0$ with constant coefficients. For the norm of the difference of resolvents, an estimate of order $\varepsilon$ is obtained.

Keywords: periodic differential operators, averaging, homogenization, threshold effect, operators of high order.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:5, 751–775

Bibliographic databases:

Document Type: Article
Received: 28.01.2010

Citation: N. A. Veniaminov, “Homogenization of periodic differential operators of high order”, Algebra i Analiz, 22:5 (2010), 69–103; St. Petersburg Math. J., 22:5 (2011), 751–775

Citation in format AMSBIB
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\paper Homogenization of periodic differential operators of high order
\jour Algebra i Analiz
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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. A. A. Kukushkin, T. A. Suslina, “Homogenization of high order elliptic operators with periodic coefficients”, St. Petersburg Math. J., 28:1 (2017), 65–108  mathnet  crossref  mathscinet  isi  elib
    2. S. E. Pastukhova, “Homogenization estimates of operator type for fourth order elliptic equations”, St. Petersburg Math. J., 28:2 (2017), 273–289  mathnet  crossref  mathscinet  isi  elib
    3. Pastukhova S.E., “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7, SI (2016), 1449–1466  crossref  mathscinet  zmath  isi  elib  scopus
    4. 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
    5. Suslina T.A., “Homogenization of the Neumann Problem For Higher Order Elliptic Equations With Periodic Coefficients”, Complex Var. Elliptic Equ., 63:7-8, SI (2018), 1185–1215  crossref  mathscinet  zmath  isi  scopus
    6. Suslina T.A., “Homogenization of Higher-Order Parabolic Systems in a Bounded Domain”, Appl. Anal., 98:1-2, SI (2019), 3–31  crossref  mathscinet  isi
  • Алгебра и анализ St. Petersburg Mathematical Journal
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