A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions
We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalue...
Saved in:
| Date: | 1999 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4748 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510912786464768 |
|---|---|
| author | Lavrent'ev, A. S. Лаврентьєв, А. С. |
| author_facet | Lavrent'ev, A. S. Лаврентьєв, А. С. |
| author_sort | Lavrent'ev, A. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:13:14Z |
| description | We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as $ε → 0$. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as $ε → 0$ and associated with natural forms of vibrations localized in the domain of perturbation of density. |
| first_indexed | 2026-03-24T03:04:32Z |
| format | Article |
| fulltext |
0026
0027
0028
0029
0030
0031
0032
0033
0034
|
| id | umjimathkievua-article-4748 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:32Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6b/bcaa0fed1978bb072983a555f31d776b.pdf |
| spelling | umjimathkievua-article-47482020-03-18T21:13:14Z A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions Сингулярно збурена спектральна задача для бігармонічного оператора з умовами Неймана Lavrent'ev, A. S. Лаврентьєв, А. С. We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as $ε → 0$. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as $ε → 0$ and associated with natural forms of vibrations localized in the domain of perturbation of density. Вивчено математичну модель композитної пластини, яка складається з двох компонент, що мають подібні пружні властивості, але відрізняються розподілом густини. Площа області, яку займає одна з компонент, є безмежно малою при $ε → 0$. Досліджується асимптотична поведінка при $ε → 0$ власних значень і власних функцій крайової задачі для бігармоиїчиого оператора з умовами Неймана. Описано чотири різні випадки граничної поведінки спектра в залежності від співвідношення густин компонент середовища. Зокрема, описано так званий ефект локальних коливань Е. Санчес-Паленсія: коливна система має зліченну серію нескінченно малих при $ε → 0$ власних частот, яким відповідають власні форми коливань, локалізовані в області збурення густини. Institute of Mathematics, NAS of Ukraine 1999-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4748 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 11 (1999); 1467–1475 Український математичний журнал; Том 51 № 11 (1999); 1467–1475 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4748/6197 https://umj.imath.kiev.ua/index.php/umj/article/view/4748/6198 Copyright (c) 1999 Lavrent'ev A. S. |
| spellingShingle | Lavrent'ev, A. S. Лаврентьєв, А. С. A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title | A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title_alt | Сингулярно збурена спектральна задача для бігармонічного оператора з умовами Неймана |
| title_full | A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title_fullStr | A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title_full_unstemmed | A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title_short | A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions |
| title_sort | singularly perturbed spectral problem for a biharmonic operator with neumann conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4748 |
| work_keys_str_mv | AT lavrent039evas asingularlyperturbedspectralproblemforabiharmonicoperatorwithneumannconditions AT lavrentʹêvas asingularlyperturbedspectralproblemforabiharmonicoperatorwithneumannconditions AT lavrent039evas singulârnozburenaspektralʹnazadačadlâbígarmoníčnogooperatorazumovaminejmana AT lavrentʹêvas singulârnozburenaspektralʹnazadačadlâbígarmoníčnogooperatorazumovaminejmana AT lavrent039evas singularlyperturbedspectralproblemforabiharmonicoperatorwithneumannconditions AT lavrentʹêvas singularlyperturbedspectralproblemforabiharmonicoperatorwithneumannconditions |