The resolvent kernel for PCF self-similar fractals

Details The resolvent kernel for PCF self-similar fractals The resolvent kernel for PCF self-similar fractals

2008-11-26

arxiv.org/abs/0811.4203v2

Trans. Amer. Math. Soc. (8) 362 (2010), 4451-4479

Mathematics

Analysis of PDEs

27 pages, 8 figures

Scientific paper

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 $SG_3$.

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