Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications
We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2013/693529 |
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| _version_ | 1849469189837815808 |
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| author | Brahim Amaziane Leonid Pankratov |
| author_facet | Brahim Amaziane Leonid Pankratov |
| author_sort | Brahim Amaziane |
| collection | DOAJ |
| description | We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter
characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed. |
| format | Article |
| id | doaj-art-fb90ea86b8f24d43867625d8a5bd12d3 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-fb90ea86b8f24d43867625d8a5bd12d32025-08-20T03:25:34ZengWileyInternational Journal of Differential Equations1687-96431687-96512013-01-01201310.1155/2013/693529693529Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and ApplicationsBrahim Amaziane0Leonid Pankratov1Laboratoire de Mathėmatiques et de leurs Applications, CNRS-UMR 5142, UNIV Pau & Pays Adour, Avenue de l’Universitė, 64000 Pau, FranceLaboratoire de Mathėmatiques et de leurs Applications, CNRS-UMR 5142, UNIV Pau & Pays Adour, Avenue de l’Universitė, 64000 Pau, FranceWe review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.http://dx.doi.org/10.1155/2013/693529 |
| spellingShingle | Brahim Amaziane Leonid Pankratov Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications International Journal of Differential Equations |
| title | Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications |
| title_full | Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications |
| title_fullStr | Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications |
| title_full_unstemmed | Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications |
| title_short | Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications |
| title_sort | homogenization in sobolev spaces with nonstandard growth brief review of methods and applications |
| url | http://dx.doi.org/10.1155/2013/693529 |
| work_keys_str_mv | AT brahimamaziane homogenizationinsobolevspaceswithnonstandardgrowthbriefreviewofmethodsandapplications AT leonidpankratov homogenizationinsobolevspaceswithnonstandardgrowthbriefreviewofmethodsandapplications |