### Abstract

Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L
_{2}
containing degrees of distance r = |x| as weight multipliers. For the Neumann conditions (stresses ere prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the "elastic" Neumann problem are presented. These results are based on the weighted Korn inequality.

Original language | English |
---|---|

Pages (from-to) | 2292-2318 |

Number of pages | 27 |

Journal | Journal of Mathematical Sciences |

Volume | 80 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 1996 |

### Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

### Cite this

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**Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains.** / Nazarov, S. A.; Slutskiǐ, A. S.

Research output › › peer-review

TY - JOUR

T1 - Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains

AU - Nazarov, S. A.

AU - Slutskiǐ, A. S.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L 2 containing degrees of distance r = |x| as weight multipliers. For the Neumann conditions (stresses ere prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the "elastic" Neumann problem are presented. These results are based on the weighted Korn inequality.

AB - Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L 2 containing degrees of distance r = |x| as weight multipliers. For the Neumann conditions (stresses ere prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the "elastic" Neumann problem are presented. These results are based on the weighted Korn inequality.

UR - http://www.scopus.com/inward/record.url?scp=53349118632&partnerID=8YFLogxK

U2 - 10.1007/BF02362388

DO - 10.1007/BF02362388

M3 - Article

VL - 80

SP - 2292

EP - 2318

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -