Expansion formulae for the homogenized determinant of anisotropic checkerboards
In this paper, some effective properties of anisotropic four-phase periodic checkerboards are studied in two-dimensional electrostatics. An explicit low-contrast second-order expansion for the determinant of the effective conductivity is given. In the case of a two-phase checkerboard with commuting...
| Main Authors: | , |
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| Format: | Journal article |
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Royal Society
2006
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| _version_ | 1797060868293263360 |
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| author | Capdeboscq, Y Briane, M |
| author_facet | Capdeboscq, Y Briane, M |
| author_sort | Capdeboscq, Y |
| collection | OXFORD |
| description | In this paper, some effective properties of anisotropic four-phase periodic checkerboards are studied in two-dimensional electrostatics. An explicit low-contrast second-order expansion for the determinant of the effective conductivity is given. In the case of a two-phase checkerboard with commuting conductivities, the expansion reduces to an explicit formula for the effective determinant (valid for any contrast) as soon as the second-order term vanishes. Such an explicit formula cannot be extended to four-phase checkerboards. A counter-example with high-contrast conductivities is provided. The construction of the counter-example is based on a factorization principle, due to Astala and Nesi, which allows us to pass from an anisotropic four-phase square checkerboard to an isotropic one with the same effective determinant. Copyright 2006 The Royal Society. |
| first_indexed | 2024-03-06T20:23:01Z |
| format | Journal article |
| id | oxford-uuid:2e77c1dc-81cb-44fc-8d30-dc8a331f563d |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:23:01Z |
| publishDate | 2006 |
| publisher | Royal Society |
| record_format | dspace |
| spelling | oxford-uuid:2e77c1dc-81cb-44fc-8d30-dc8a331f563d2022-03-26T12:49:05ZExpansion formulae for the homogenized determinant of anisotropic checkerboardsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2e77c1dc-81cb-44fc-8d30-dc8a331f563dSymplectic Elements at OxfordRoyal Society2006Capdeboscq, YBriane, MIn this paper, some effective properties of anisotropic four-phase periodic checkerboards are studied in two-dimensional electrostatics. An explicit low-contrast second-order expansion for the determinant of the effective conductivity is given. In the case of a two-phase checkerboard with commuting conductivities, the expansion reduces to an explicit formula for the effective determinant (valid for any contrast) as soon as the second-order term vanishes. Such an explicit formula cannot be extended to four-phase checkerboards. A counter-example with high-contrast conductivities is provided. The construction of the counter-example is based on a factorization principle, due to Astala and Nesi, which allows us to pass from an anisotropic four-phase square checkerboard to an isotropic one with the same effective determinant. Copyright 2006 The Royal Society. |
| spellingShingle | Capdeboscq, Y Briane, M Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title | Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title_full | Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title_fullStr | Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title_full_unstemmed | Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title_short | Expansion formulae for the homogenized determinant of anisotropic checkerboards |
| title_sort | expansion formulae for the homogenized determinant of anisotropic checkerboards |
| work_keys_str_mv | AT capdeboscqy expansionformulaeforthehomogenizeddeterminantofanisotropiccheckerboards AT brianem expansionformulaeforthehomogenizeddeterminantofanisotropiccheckerboards |