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...
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| Format: | Journal article |
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Royal Society
2006
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| Summary: | 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. |
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