Energy and Laplacian of Fractal Interpolation Functions

Xiao-Hui Li; Huo-Jun Ruan

paper

Energy and Laplacian of Fractal Interpolation Functions

arXiv:1607.06176

Abstract

In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is an FIF with uniform vertical scaling factor $\frac{1}{5}$ : $Δ u = 0$ on $S G ∖ {q_{1}, q_{2}, q_{3}}$ , and $u (q_{i}) = a_{i}$ , $i = 1, 2, 3$ , where $q_{i}$ , $i = 1, 2, 3$ , are boundary points of SG.

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