### 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 language | English |
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Pages (from-to) | 1385-1422 |

Number of pages | 38 |

Journal | Sbornik Mathematics |

Volume | 189 |

Issue number | 9-10 |

DOIs | |

Publication status | Published - 1 Jan 1998 |

### Scopus subject areas

- Algebra and Number Theory

### Cite this

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**Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity.** / Nazarov, S. A.; Slutskiǐ, A.

Research output › › peer-review

TY - JOUR

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.

PY - 1998/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=0038911671&partnerID=8YFLogxK

U2 - 10.1070/SM1998v189n09ABEH000353

DO - 10.1070/SM1998v189n09ABEH000353

M3 - Article

VL - 189

SP - 1385

EP - 1422

JO - Sbornik Mathematics

JF - Sbornik Mathematics

SN - 1064-5616

IS - 9-10

ER -