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Article overview
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The resolvent kernel for PCF self-similar fractals | | Marius Ionescu
; Erin P. J. Pearse
; Luke G. Rogers
; Huo-Jun Ruan
; Robert S. Strichartz
; | | Date: |
26 Nov 2008 | | 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 $(lambda
mathbb{I} - Delta)^{-1} f(x) = int G^{(lambda)}(x,y) f(y) dmu(y)$. 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 $SG_3$. | | Source: | arXiv, 0811.4203 | | Services: | Forum | Review | PDF | Favorites | |
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