The resolvent kernel for PCF self-similar fractals

Abstract

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 $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} 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. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$.