PCF自相似分形的预解核
The resolvent kernel for PCF self-similar fractals
摘要：
对于定义在p.c.f.上的拉普拉斯算子$Δ$，在自相似分形的基础上，给出了Dirichlet边界条件和Neumann边界条件下Laplacian算子的预解核的显式表达式.也就是说，我们构造一个对称函数$G^{(λ)}$，它求解$(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$。该方法类似于Kigami在\cite [ \S 3.5]{Kig 01}中构建的绿核，并表示为某个函数$ψ^{(λ)}$的缩放和“平移”副本的总和，该函数可被视为预解方程的基本解。给出了单位区间、标准Sierpinski垫片和3层Sierpinski垫片$SG_3$的显式预解核公式的例子.
英文摘要：
For the Laplacian $Δ$ 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 $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. 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 $ψ^{(λ)}$ 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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