Eikonal equations on the Sierpinski gasket

TL;DR

This paper investigates the eikonal equation on the Sierpinski gasket by analyzing graph-based solutions on prefractals and demonstrating their convergence to a unique metric viscosity solution on the fractal.

Contribution

It introduces a method to define and analyze the eikonal equation on fractals via graph approximations and characterizes the limit as a unique viscosity solution.

Findings

Graph eikonal solutions converge to a well-defined limit

The limit function is characterized as a unique metric viscosity solution

Method extends PDE analysis to fractal geometries

Abstract

We study the eikonal equation on the Sierpinski gasket in the spirit of the construction of the Laplacian in Kigami [8]: we consider graph eikonal equations on the prefractals and we show that the solutions of these problems converge to a function defined on the fractal set. We characterize this limit function as the unique metric viscosity solution to the eikonal equation on the Sierpinski gasket according to the definition introduced in [3].