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Nonlocal Effects in Two-Dimensional Conductivity

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Abstract

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a ε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L 1-boundedness of the sequence a ε , the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness and L 1-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.

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  1. Centre de Mathématiques, I.N.S.A. de Rennes & I.R.M.A.R, 20 avenue des Buttes de Coësmes, CS 14315 - 35043, Rennes Cedex, France

    Marc Briane

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  1. Marc Briane
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Correspondence to Marc Briane.

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Communicated by S. Müller

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Briane, M. Nonlocal Effects in Two-Dimensional Conductivity. Arch Rational Mech Anal 182, 255–267 (2006). https://doi.org/10.1007/s00205-006-0427-4

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  • DOI: https://doi.org/10.1007/s00205-006-0427-4

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