The resolvent kernel for PCF self-similar 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 and $G^{(\lambda )}$ 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 $SG_3$.