The Resolvent Kernel for Pcf Self-similar Fractals

For the Laplacian ∆ 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 − ∆)−1 f (x) = ∫ G(λ)(x, y) f (y) dμ(y). The method is similar to Kigami’s construction of the Green kernel in [Kig01, §3.5] 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 S G3.