The Laplacian on p.c.f. self-similar sets via the method of averages

We show how the symmetric Laplacian on p.c.f. self-similar sets, together with its associated Dirichlet form and harmonic functions, can be defined entirely in terms of average values of a function over basic sets. The approach combined the constructive limit-of-difference-quotients method of Kigami and the method of averages introduced by Kusuoka and Zhou for the Sierpinski carpet. We consider well-known examples, such as the unit interval, the Vicsek set,the hexagasket, and SG(4). This paper has generalized the results in [11, 13, 14], but a different proof is needed. (C) 2011 Elsevier Ltd. All rights reserved.