Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet

TL;DR

This paper explores harmonic functions and the Laplacian spectrum on the Sierpinski carpet, providing computational data and insights into boundary value problems and eigenfunctions.

Contribution

It implements an algorithm based on Kusuoka and Zhou's method to approximate solutions, generating extensive data on harmonic and eigenfunctions on the fractal.

Findings

Data on harmonic functions with prescribed boundary values

Eigenfunctions of the Laplacian under different boundary conditions

Ideas for defining normal derivatives on the Sierpinski carpet boundary

Abstract

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.