p-energy and p-harmonic functions on Sierpinski gasket type fractals

摘要

We show that it is possible to define a notion of p-energy for functions defined on a class of fractals including the Sierpinski gasket (SG) for any value of p, 1 < p < infinity, extending the construction of Kigami for p = 2, as a renormalized limit of modified p-energies on a sequence of graphs. Our proof is non-constructive, and does not settle the question of uniqueness. Based on the p-energy we may define p-harmonic functions as p-energy minimizers subject to boundary conditions, but again uniqueness is only conjectural. We present some numerical data as a complement to our results. This work is intended to pave the way for an eventual theory of p-Laplacians on fractals.