RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (3416 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{Naz91}
\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
\mathnet{http://mi.mathnet.ru/msb1319}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1124104}
\zmath{https://zbmath.org/?q=an:0782.35005}
\transl
\jour Math. USSR-Sb.
\yr 1992
\vol 73
\issue 1
\pages 79--110
\crossref{https://doi.org/10.1070/SM1992v073n01ABEH002536}

• http://mi.mathnet.ru/eng/msb1319
• http://mi.mathnet.ru/eng/msb/v182/i5/p692

 SHARE:

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
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
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
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
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
6. Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S161–S167
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
8. Panasenko G., “The Partial Homogenization: Continuous and Semi-Discretized Versions”, Math. Models Meth. Appl. Sci., 17:8 (2007), 1183–1209
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
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
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
12. Malakhova I.S., “Kraevaya zadacha dlya ellipticheskogo uravneniya s bystroostsilliruyuschimi koeffitsientami v trekhmernom sluchae”, Vestnik Chelyabinskogo gosudarstvennogo universiteta, 2011, no. 27, 85–93
13. I. S. Malakhova, “Kraevaya zadacha dlya ellipticheskogo uravneniya s bystroostsilliruyuschimi koeffitsientami v trekhmernom sluchae”, Vestnik ChelGU, 2011, no. 14, 85–93
14. Christophe Prange, “Asymptotic Analysis of Boundary Layer Correctors in Periodic Homogenization”, SIAM J. Math. Anal, 45:1 (2013), 345
15. S. A. Nazarov, “Osrednenie plastin Kirkhgofa, soedinennykh zaklepkami, kotorye modeliruyutsya tochechnymi usloviyami Soboleva”, Algebra i analiz, 32:2 (2020), 143–200
•  Number of views: This page: 550 Full text: 116 References: 45 First page: 3