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 $μ$ we show there is a full $μ$-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.
It’s been two years since I last worked on this project, but I finally have my first preprint out! This was from an REU I did in 2017, and it was an absolute blast. Analysis on fractals is an incredible field that combines various approaches to develop a theory of analysis on self-similar spaces that lack the nice properties of smoothness but allow for a lot of interesting math.