Weak Homogenization of Anisotropic Diffusion on Pre-Sierpiński Carpets

Abstract:

We study a kind of “restoration of isotropy” on the pre-Sierpiński. Let and be the effective resistances in the x and y directions, respectively, of the Sierpiński at the n ^th 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.