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Mat. Sb., 1998, Volume 189, Number 9, Pages 107–142 (Mi msb353)  

This article is cited in 2 scientific papers (total in 2 papers)

Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity

S. A. Nazarova, A. S. Slutskijb

a Saint-Petersburg State University
b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: Asymptotic representations of the solutions of boundary-value problems for a second-order equation with rapidly oscillating coefficients in a domain with a small cavity (of diameter comparable with the period of oscillation) are found and substantiated. Dirichlet or Neumann conditions are set at the boundary of the domain. In addition to an asymptotic series of structure standard for homogenization theory there occur terms describing the boundary layer phenomenon near the opening, while the solutions of the homogenized problem and their rapidly oscillating correctors acquire singularities at the contraction point of the openings. The dimension of the domain and some other factors influence even the leading term of the asymptotic formula. Some generalizations, including ones to the system of elasticity theory, are discussed.

DOI: https://doi.org/10.4213/sm353

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English version:
Sbornik: Mathematics, 1998, 189:9, 1385–1422

Bibliographic databases:

UDC: 517.946
MSC: Primary 35B25, 35B27, 35C20; Secondary 35J25
Received: 16.12.1996 and 15.06.1998

Citation: 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”, Mat. Sb., 189:9 (1998), 107–142; Sb. Math., 189:9 (1998), 1385–1422

Citation in format AMSBIB
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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. 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
    2. 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  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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