The Laplacian on the unit square in a self-similar manner

TL;DR

This paper constructs the standard Laplacian on the unit square using a self-similar approach, combining methods from Kigami and Kusuoka-Zhou to understand harmonic functions and resistance forms.

Contribution

It introduces a novel self-similar construction of the Laplacian on the unit square, blending Kigami's difference-quotients with Kusuoka-Zhou's averaging methods.

Findings

Explicit self-similar resistance form derived

Laplacian constructed via limit-of-difference-quotients

Harmonic functions characterized on the square

Abstract

In this paper, we show how to construct the standard Laplacian on the unit square in a self-similar manner. We rewrite the familiar mean value property of planar harmonic functions in terms of averge values on small squares, from which we could know how the planar self-similar resistance form and the Laplacian look like. This approach combines the constructive limit-of-difference-quotients method of Kigami for p.c.f. self-similar sets and the method of averages introduced by Kusuoka and Zhou for the Sierpinski carpet.