Absolutely Minimizing Lipschitz Extensions and Infinity Harmonic Functions on the Sierpinski gasket

TL;DR

This paper investigates the infinity Laplace operator and Absolutely Minimizing Lipschitz Extensions on the Sierpinski gasket, extending Kigami's classical Laplacian construction to fractal settings.

Contribution

It introduces a notion of infinity harmonic functions on pre-fractal sets and proves their convergence to solutions of the Lipschitz extension problem on the Sierpinski gasket.

Findings

Infinity harmonic functions solve a Lipschitz extension problem in the discrete setting.

Limit of these functions solves the Absolutely Minimizing Lipschitz Extension problem on the gasket.

Provides a fractal analogue of classical harmonic analysis methods.

Abstract

Aim of this note is to study the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket in the spirit of the classical construction of Kigami for the Laplacian. We introduce a notion of infinity harmonic functions on pre-fractal sets and we show that these functions solve a Lipschitz extension problem in the discrete setting. Then we prove that the limit of the infinity harmonic functions on the pre-fractal sets solves the Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket.