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Weak Homogenization of Anisotropic Diffusion on Pre-Sierpiński Carpets |
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| Authors: | Martin T Barlow Kumiko Hattori Tetsuya Hattori Hiroshi Watanabe |
| Institution: | Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada. E-mail: barlow@math.ubc.ca, CA Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 113, Japan.?E-mail: f36320@m-unix.cc.u-tokyo.ac.jp, JP Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Tokyo 171, Japan.?E-mail: d33925@m-unix.cc.u-tokyo.ac.jp, JP Department of Mathematics, Nippon Medical School, Kosugi, Nakahara Kawasaki 211, Japan.?E-mail: d34335@m-unix.cc.u-tokyo.ac.jp, JP |
| 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. Received: 26 June 1996 / Accepted: 25 November 1996 |
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