Energy and Laplacian of Fractal Interpolation Functions

TL;DR

This paper investigates the properties of fractal interpolation functions on self-similar sets, analyzes their Laplacian on the Sierpinski gasket, and applies these findings to solve a Dirichlet problem with FIF solutions.

Contribution

It characterizes the finiteness of FIFs on self-similar sets and links their Laplacian to solutions of boundary value problems on fractals.

Findings

FIFs are finite on certain self-similar sets.

Laplacian of FIFs on Sierpinski gasket is studied.

A Dirichlet problem solution is shown to be an FIF with specific scaling.

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 $\frac{1}{5}$: $\Delta u=0$ on $SG\setminus \{q_1,q_2,q_3\}$, and $u(q_i)=a_i$, $i=1,2,3$, where $q_i$, $i=1,2,3$, are boundary points of SG.