Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket

摘要

We consider the energy Laplacian Δ_v on the Sierpinski gasket (SG), which is defined by the standard self-similar energy ε and the Kusuoka measure, and is different from the standard self-similar Laplacian A. We study the local behavior of function u ∈ dom Δ_v~k near a boundary point q_0. We define jets of local derivatives at q_0 and estimate the decay rate of u near q_0 in terms of the vanishing of jet. This can be used to define Taylor approximating polynomials with error estimates. Analogous results are known for the standard Laplacian, but the results here are quite different. We also confirm experimentally the absolute continuity of different energy measures, and give experimental evidence that the density is p-integrable for p < (log(15))/(log(9)).