Harmonic Gradients on Higher Dimensional Sierpinski Gaskets

Abstract

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $S{G}_N$, $N>3$, proving results that generalize those of Teplyaev. When $S{G}_N$ 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 $S{G}_N$. 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.

9 pages