Homogenization of periodic systems with large potentials
We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by (Epsilon) the period, the potential is scaled as (Epsilon. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution ca...
| Main Authors: | , , , |
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| Format: | Journal article |
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Springer-Verlag
2004
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| _version_ | 1797069365578825728 |
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| author | Capdeboscq, Y Piatnitski, A Allaire, G Siess, V |
| author_facet | Capdeboscq, Y Piatnitski, A Allaire, G Siess, V |
| author_sort | Capdeboscq, Y |
| collection | OXFORD |
| description | We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by (Epsilon) the period, the potential is scaled as (Epsilon. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition. |
| first_indexed | 2024-03-06T22:23:21Z |
| format | Journal article |
| id | oxford-uuid:55cef9b2-1aa4-42ad-a70d-7fe60fa92ad2 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:23:21Z |
| publishDate | 2004 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | oxford-uuid:55cef9b2-1aa4-42ad-a70d-7fe60fa92ad22022-03-26T16:46:34ZHomogenization of periodic systems with large potentialsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:55cef9b2-1aa4-42ad-a70d-7fe60fa92ad2Symplectic Elements at OxfordSpringer-Verlag2004Capdeboscq, YPiatnitski, AAllaire, GSiess, VWe consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by (Epsilon) the period, the potential is scaled as (Epsilon. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition. |
| spellingShingle | Capdeboscq, Y Piatnitski, A Allaire, G Siess, V Homogenization of periodic systems with large potentials |
| title | Homogenization of periodic systems with large potentials |
| title_full | Homogenization of periodic systems with large potentials |
| title_fullStr | Homogenization of periodic systems with large potentials |
| title_full_unstemmed | Homogenization of periodic systems with large potentials |
| title_short | Homogenization of periodic systems with large potentials |
| title_sort | homogenization of periodic systems with large potentials |
| work_keys_str_mv | AT capdeboscqy homogenizationofperiodicsystemswithlargepotentials AT piatnitskia homogenizationofperiodicsystemswithlargepotentials AT allaireg homogenizationofperiodicsystemswithlargepotentials AT siessv homogenizationofperiodicsystemswithlargepotentials |