Duality results in the homogenization of two-dimensional high-contrast conductivities

Marc Briane; David Manceau; Marc Briane; David Manceau

doi:10.3934/nhm.2008.3.509

Networks and Heterogeneous Media

2008, Volume 3, Issue 3: 509-522. doi: 10.3934/nhm.2008.3.509

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Duality results in the homogenization of two-dimensional high-contrast conductivities

Marc Briane ^{1
,},
David Manceau ^{2
,}

1.
Centre de Mathématiques INSA de Rennes & IRMAR, 20 ave. des Buttes de Coësmes, 35043 Rennes Cedex
2.
I.R.M.A.R., Université de Rennes 2, Rennes Cedex

Received: 01 January 2008
Primary: 35J25, 35B27; Secondary: 35B25.

The paper deals with some extensions of the Keller-Dykhneduality relations arising in the classical homogenization of two-dimensional uniformly bounded conductivities, to the case of high-contrast conductivities. Only assuming a $L^1$-bound on the conductivity we prove that the conductivity and its dual converge respectively, in a suitable sense, to the homogenized conductivity and its dual. In the periodic case a similar duality result is obtained under a less restrictive assumption.
- Homogenization,
- duality in two-dimensional media,
- high and low conductivities,
- periodic microstructures
Citation: Marc Briane, David Manceau. Duality results in the homogenization of two-dimensional high-contrast conductivities[J]. Networks and Heterogeneous Media, 2008, 3(3): 509-522. doi: 10.3934/nhm.2008.3.509

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Abstract

The paper deals with some extensions of the Keller-Dykhneduality relations arising in the classical homogenization of two-dimensional uniformly bounded conductivities, to the case of high-contrast conductivities. Only assuming a $L^1$-bound on the conductivity we prove that the conductivity and its dual converge respectively, in a suitable sense, to the homogenized conductivity and its dual. In the periodic case a similar duality result is obtained under a less restrictive assumption.
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