Weak Homogenization of Anisotropic Diffusion on Pre-Sierpi?ski Carpets

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We study a kind of “restoration of isotropy” on the pre-Sierpinski. Let \(\) and \(\) be the effective resistances in the x and y directions, respectively, of the Sierpinski at the nth stage of its construction, if it is made of anisotropic material whose anisotropy is parametrized by the ratio of resistances for a unit square: \(\). We prove that isotropy is weakly restored asymptotically in the sense that for all sufficiently large n the ratio \(\) is bounded by positive constants independent of r. The ratio decays exponentially fast when r≫ 1. Furthermore, it is proved that the effective resistances asymptotically grow exponentially with an exponent equal to that found by Barlow and Bass for the isotropic case r = 1.

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