What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals

Abstract

We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and Δf is defined as a continuous function. We show that if f is a nonconstant function in the domain of Δ, then f2 is not in the domain of Δ. We give two proofs of this fact. The first is based on the analog of the pointwise identity Δf2−2fΔf=|∇f|2, where we show that |∇f|2 does not exist as a continuous function. In fact the correct interpretation of Δf2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of Δ, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for |f(x)−f(x0)| in terms of d(x, x0)β for a certain value β, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of Δ.