The resolvent kernel for PCF self-similar fractals

by Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, Robert S. Strichartz

For the Laplacian Δ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function G(λ) which solves (λI−Δ)−1f(x)=∫G(λ)(x,y)f(y)dμ(y). The method is similar to Kigami’s construction of the Green kernel and is expressed as a sum of scaled and “translated” copies of a certain function ψ(λ) which may be considered as a fundamental solution of the resolvent equation.

Arxiv (preprint) version

Published version in Trans AMS