Nonlocal Effects in Two-Dimensional Conductivity

The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity aε is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L1-boundedness of the sequence aε, 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 L1-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions.