The resolvent kernel for PCF self-similar fractals
2010; American Mathematical Society; Volume: 362; Issue: 8 Linguagem: Inglês
10.1090/s0002-9947-10-05098-1
ISSN1088-6850
AutoresMarius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, Robert S. Strichartz,
Tópico(s)Advanced Mathematical Theories and Applications
ResumoFor the Laplacian Δ \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 ( λ ) G^{(\lambda )} which solves ( λ I − Δ ) − 1 f ( x ) = ∫ G ( λ ) ( x , y ) f ( y ) d μ ( y ) (\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 ( λ ) G^{(\lambda )} is expressed as a sum of scaled and “translated” copies of a certain function ψ ( λ ) \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 S G 3 SG_3 .
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