The resolvent kernel for PCF self-similar fractals

TL;DR

This paper derives explicit formulas for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals, providing a fundamental solution approach for Dirichlet and Neumann boundary conditions.

Contribution

It introduces a method to explicitly construct the resolvent kernel on p.c.f. fractals using scaled and translated fundamental solutions.

Findings

Explicit resolvent kernel formulas for the unit interval and Sierpinski gaskets.

Method generalizes Kigami's Green kernel construction.

Examples demonstrate the applicability to various fractals.

Abstract

For the Laplacian $\Delta$ 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 $G^{(\lambda)}$ which solves $(\lambda \mathbb{I} - \Delta)^{-1} f(x) = \int G^{(\lambda)}(x,y) f(y) d\mu(y)$. 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 $\psi^{(\lambda)}$ 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 $SG_3$.