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

Research outputpeer-review

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.

Original languageEnglish
Pages (from-to)1385-1422
Number of pages38
JournalSbornik Mathematics
Volume189
Issue number9-10
DOIs
Publication statusPublished - 1 Jan 1998

Scopus subject areas

  • Algebra and Number Theory

Cite this

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title = "Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity",
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.",
author = "Nazarov, {S. A.} and A. Slutskiǐ",
year = "1998",
month = "1",
day = "1",
doi = "10.1070/SM1998v189n09ABEH000353",
language = "English",
volume = "189",
pages = "1385--1422",
journal = "Sbornik Mathematics",
issn = "1064-5616",
publisher = "Turpion Ltd.",
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T1 - Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity

AU - Nazarov, S. A.

AU - Slutskiǐ, A.

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Y1 - 1998/1/1

N2 - 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.

AB - 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.

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DO - 10.1070/SM1998v189n09ABEH000353

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JF - Sbornik Mathematics

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ER -