The Laplacian on the unit square in a self-similar manner
  In this paper, we show how to construct the standard Laplacian on the unit square in a self-similar manner. We rewrite the familiar mean value property of planar harmonic functions in terms of averge values on small squares, from which we could know how the planar self-similar resistance form and the Laplacian look like. This approach combines the constructive limit-of-difference-quotients method of Kigami for p.c.f. self-similar sets and the method of averages introduced by Kusuoka and Zhou for the Sierpinski carpet. 
Hua Qiu and Haoran Tian
math.FA