Group Members
Luke Brown, Giovanni E Ferrer Suarez, Karuna Sangam.
Supervisors
Gamal Mograby, Dan Kelleher, Luke Rogers, Sasha Teplyaev.
Overview
Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.