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Mat. Sb., 1991, Volume 182, Number 5, Pages 692–722 (Mi msb1319)  

This article is cited in 14 scientific papers (total in 14 papers)

Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle

S. A. Nazarov


Abstract: A complete asymptotic expansion is found for the solution of the Dirichlet problem for a second-order scalar equation in a rectangle. The exponents of the powers of $\varepsilon$ in the series are (generally speaking, nonintegral) nonnegative numbers of the form $p+q_1\alpha_1\pi^{-1}+…+q_4\alpha_4\pi^{-1}$, where $p$, $q_j=0,1,…$, and $\alpha_j$ is the opening of the angle which is transformed into a quarter plane under the change of coordinates taking the Laplace operator into the principal part of the averaged operator at the vertex $O_j$ of the rectangle. The coefficients of the series for rational $\alpha_j\pi^-1$ may depend in polynomial fashion on $\log\varepsilon$. It is shown that the algorithm also does not change in the case of a system of differential equations or in the case of a domain bounded by polygonal lines with vertices at the nodes of an $\varepsilon$-lattice. The spectral problem is considered; asymptotic formulas for the eigenvalue $\lambda(\varepsilon)$ and the eigenfunction are obtained under the assumption that $\lambda(0)$ is a simple eigenvalue of the averaged Dirichlet problem.

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English version:
Mathematics of the USSR-Sbornik, 1992, 73:1, 79–110

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J25, 35C10; Secondary 35P15
Received: 17.04.1990

Citation: S. A. Nazarov, “Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle”, Mat. Sb., 182:5 (1991), 692–722; Math. USSR-Sb., 73:1 (1992), 79–110

Citation in format AMSBIB
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\by S.~A.~Nazarov
\paper Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a~rectangle
\jour Mat. Sb.
\yr 1991
\vol 182
\issue 5
\pages 692--722
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..73...79N}
\transl
\jour Math. USSR-Sb.
\yr 1992
\vol 73
\issue 1
\pages 79--110
\crossref{https://doi.org/10.1070/SM1992v073n01ABEH002536}
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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. Nazarov S., “Asymptotics at Infinity of the Solution to the Dirichlet Problem for a System of Equations with Periodic Coefficients in an Angular Domain”, Russ. J. Math. Phys., 3:3 (1995), 297–326  mathscinet  zmath  isi
    2. F Blanc, S.A Nazarov, “Asymptotics of solutions to the Poisson problem in a perforated domain with corners”, Journal de Mathématiques Pures et Appliquées, 76:10 (1997), 893  crossref  elib
    3. S. A. Nazarov, A. S. Slutskij, “Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity”, Sb. Math., 189:9 (1998), 1385–1422  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Gadyl'shin R., “Asymptotics of the Eigenvalues of a Boundary Value Problem with Rapidly Oscillating Boundary Conditions”, Differ. Equ., 35:4 (1999), 540–551  mathnet  mathscinet  isi
    5. Teplinskii A., “Asymptotic Expansions for the Eigenvalues and the Eigenfunctions of Boundary Value Problems with Rapidly Oscillating Coefficients in a Layer”, Differ. Equ., 36:6 (2000), 911–917  mathnet  crossref  mathscinet  isi
    6. Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S161–S167  mathnet  mathscinet  zmath  elib
    7. S. A. Nazarov, “Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^\infty$ estimates for asymptotic remainders”, St. Petersburg Math. J., 18:2 (2007), 269–304  mathnet  crossref  mathscinet  zmath  elib
    8. Panasenko G., “The Partial Homogenization: Continuous and Semi-Discretized Versions”, Math. Models Meth. Appl. Sci., 17:8 (2007), 1183–1209  crossref  mathscinet  zmath  isi
    9. Yao Zhengan, Zhao Hongxing, “Homogenization of a Stationary Navier–Stokes Flow in Porous Medium with Thin Film”, Acta Math. Sci., 28:4 (2008), 963–974  mathscinet  zmath  isi
    10. V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983  mathnet  crossref  mathscinet  zmath  isi
    11. Cardone G., Nazarov S.A., Piatnitski A.L., “On the rate of convergence for perforated plates with a small interior Dirichlet zone”, Z Angew Math Phys, 62:3 (2011), 439–468  crossref  isi
    12. Malakhova I.S., “Kraevaya zadacha dlya ellipticheskogo uravneniya s bystroostsilliruyuschimi koeffitsientami v trekhmernom sluchae”, Vestnik Chelyabinskogo gosudarstvennogo universiteta, 2011, no. 27, 85–93  elib
    13. I. S. Malakhova, “Kraevaya zadacha dlya ellipticheskogo uravneniya s bystroostsilliruyuschimi koeffitsientami v trekhmernom sluchae”, Vestnik ChelGU, 2011, no. 14, 85–93  mathnet
    14. Christophe Prange, “Asymptotic Analysis of Boundary Layer Correctors in Periodic Homogenization”, SIAM J. Math. Anal, 45:1 (2013), 345  crossref
  • Математический сборник - 1991 Sbornik: Mathematics (from 1967)
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