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Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. (English) Zbl 1118.78009
In this paper we study the positivity of the determinant of the local electric field in a conducting composite. We know by [G. Alessandrini and V. Nesi, Univalent \(\sigma\)-harmonic mappings. Arch. Ration. Mech. Anal. 158, No. 2, 155–171 (2001; Zbl 0977.31006)] that the positivity holds true in two dimensions for any periodic structure. Using a different approach from [M. Briane and V. Nesi, ESAIM, Control Optim. Calc. Var. 10, 452–477 (2004; Zbl 1072.74057)] we prove that is also the case for a laminate microstructure in any dimension. However, and this is the main result of the paper, we provide an example of a two-phase three-dimensional periodic composite for which the determinant changes sign.

MSC:
78A48 Composite media; random media in optics and electromagnetic theory
78M40 Homogenization in optics and electromagnetic theory
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