Expansion formulae for the homogenized determinant of anisotropic checkerboards

Journal article

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.

Authors: Capdeboscq, Y; Briane, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Royal Society
• Journal: Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences
• Volume: 462
• Issue: 2073
• Pages: 2759-2779
• Publication date: 2006-09-08
• DOI: 10.1098/rspa.2006.1690
• EISSN: 1471-2946
• ISSN: 1471-2946
• Keywords: homogenization; explicit anisotropic formulae; checkerboards