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Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity

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Abstract.

In this paper we study the positivity of the determinant of the local electric field in a conducting composite. We know by [1] that the positivity holds true in two dimensions for any periodic structure. Using a different approach from [11] 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.

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Authors and Affiliations

  1. Centre de Mathématiques, INSA de Rennes & IRMAR, 20 avenue des Buttes de Coësmes, 35043, Rennes Cedex, France

    Marc Briane

  2. Department of Mathematics, The University of Utah, 155 S 1400 E Rm 233 Salt Lake City, Utah, 84112-0090, USA

    Graeme W. Milton

  3. Dipartimento di Matematica, Universitá di Roma, La Sapienza, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Vincenzo Nesi

Authors
  1. Marc Briane
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  2. Graeme W. Milton
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  3. Vincenzo Nesi
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Correspondence to Graeme W. Milton.

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Briane, M., Milton, G. & Nesi, V. Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity. Arch. Rational Mech. Anal. 173, 133–150 (2004). https://doi.org/10.1007/s00205-004-0315-8

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