Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Homogenization of materials with sign changing coefficients

Pages: 1137 – 1154



Renata Bunoiu (Institut Élie Cartan de Lorraine, Université de Lorraine and CNRS, Metz, France)

Karim Ramdani (Inria, SPHINX, Villers-lès-Nancy, France)


We investigate a periodic homogenization problem involving two isotropic materials with conductivities of different signs: a classical material and a metamaterial (or negative material). Combining the T-coercivity approach and the unfolding method for homogenization, we prove well-posedness results for the initial and the homogenized problems and we obtain a convergence result. These results are obtained under the condition that the contrast between the two conductivities is large enough in modulus. The homogenized matrix, is generally anisotropic and indefinite, but it is shown to be isotropic and (positive or negative) definite for particular geometries having symmetries.


metamaterials, homogenization, periodic unfolding, T-coercivity, indefinite operators

2010 Mathematics Subject Classification

35B27, 35M99, 35Q60, 78A48, 78M30

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Published 6 April 2016