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 Algebra i Analiz: Year: Volume: Issue: Page: Find

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

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

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Document Type: Article

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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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. A. A. Kukushkin, T. A. Suslina, “Homogenization of high order elliptic operators with periodic coefficients”, St. Petersburg Math. J., 28:1 (2017), 65–108
2. S. E. Pastukhova, “Homogenization estimates of operator type for fourth order elliptic equations”, St. Petersburg Math. J., 28:2 (2017), 273–289
3. Pastukhova S.E., “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7, SI (2016), 1449–1466
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
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
6. Suslina T.A., “Homogenization of Higher-Order Parabolic Systems in a Bounded Domain”, Appl. Anal., 98:1-2, SI (2019), 3–31
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