Harmonic Gradients on Higher Dimensional Sierpinski Gaskets

TL;DR

This paper investigates the conditions under which functions with continuous Laplacian on higher-dimensional Sierpinski Gaskets possess gradients, revealing measure-dependent differentiability properties and extending previous results.

Contribution

It generalizes differentiability criteria for functions with continuous Laplacian to higher-dimensional Sierpinski Gaskets and explores measure effects on gradient existence.

Findings

Continuity of Laplacian implies gradient exists on a full measure subset.

Standard measure on $SG_N$ does not guarantee gradient everywhere.

Certain non-uniform measures ensure the gradient exists and is continuous everywhere.

Abstract

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $\mu$ we show there is a full $\mu$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere, in sharp contrast to the case with the standard measure.