On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains
We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W m 1(Ω s ) → [W m 1(Ω s )]* in a sequence of perforated domains Ω s ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω s , we prove the following principle of compensated compactness:...
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| Date: | 2000 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4559 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510702076166144 |
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| author | Skrypnik, I. V. Скрипник, І. В. |
| author_facet | Skrypnik, I. V. Скрипник, І. В. |
| author_sort | Skrypnik, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:31:24Z |
| description | We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W m 1(Ω s ) → [W m 1(Ω s )]* in a sequence of perforated domains Ω s ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω s , we prove the following principle of compensated compactness: \({\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0\) , where r s(x) and z s(x) are sequences weakly convergent in W m 1(Ω) and such that r s(x) is an analog of a corrector for a homogenization problem and z s(x) is an arbitrary sequence from \({\mathop {W_m^1 }\limits^ \circ} (\Omega _s)\) whose weak limit is equal to zero. |
| first_indexed | 2026-03-24T03:01:11Z |
| format | Article |
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| id | umjimathkievua-article-4559 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:01:11Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/10/9f2a005194da0296e5da255d029ed910.pdf |
| spelling | umjimathkievua-article-45592020-03-18T20:31:24Z On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains Про компенсовану компактність для нелінійних еліптичних задач у перфорованих областях Skrypnik, I. V. Скрипник, І. В. We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W m 1(Ω s ) → [W m 1(Ω s )]* in a sequence of perforated domains Ω s ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω s , we prove the following principle of compensated compactness: \({\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0\) , where r s(x) and z s(x) are sequences weakly convergent in W m 1(Ω) and such that r s(x) is an analog of a corrector for a homogenization problem and z s(x) is an arbitrary sequence from \({\mathop {W_m^1 }\limits^ \circ} (\Omega _s)\) whose weak limit is equal to zero. Розглядається послідовність задач Діріхле для нелінійного дивергентного еліптичного оператора $A$: $W_m^1(Ω_s ) → [W_m^1(Ω_s )]^{*}$ в послідовності перфорованих областей $Ω_s ⊂ Ω$. За умови на локальну ємність множини $Ω \backslash Ω_s$ доведено такий принцип компенсованої компактності: ${\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0$, де $r_s(x), z_s(x)$ —слабко збіжні в послідовності такі, що $W_m^1(Ω)$ аналог коректора для задачі усереднення, $z_s (х)$ — довільна послідовність в ${\mathop {W_m^1 }\limits^ \circ} (\Omega _s)$, слабка границя якої дорівнює нулю. Institute of Mathematics, NAS of Ukraine 2000-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4559 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 11 (2000); 1534-1549 Український математичний журнал; Том 52 № 11 (2000); 1534-1549 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/4559/5822 https://umj.imath.kiev.ua/index.php/umj/article/view/4559/5823 Copyright (c) 2000 Skrypnik I. V. |
| spellingShingle | Skrypnik, I. V. Скрипник, І. В. On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title | On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title_alt | Про компенсовану компактність для нелінійних еліптичних задач у перфорованих областях |
| title_full | On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title_fullStr | On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title_full_unstemmed | On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title_short | On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains |
| title_sort | on compensated compactness for nonlinear elliptic problems in perforated domains |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4559 |
| work_keys_str_mv | AT skrypnikiv oncompensatedcompactnessfornonlinearellipticproblemsinperforateddomains AT skripnikív oncompensatedcompactnessfornonlinearellipticproblemsinperforateddomains AT skrypnikiv prokompensovanukompaktnístʹdlânelíníjnihelíptičnihzadačuperforovanihoblastâh AT skripnikív prokompensovanukompaktnístʹdlânelíníjnihelíptičnihzadačuperforovanihoblastâh |