Some Properties of Laplacians on Fractals

Author: Gross R.L.S.

Source: Journal of Functional Analysis, Volume 164, Number 2, June 1999 , pp. 181-208(28)

Publisher: Academic Press

Key:

- Free Content

- New Content

- Subscribed Content

- Free Trial Content

Abstract:

Kigami has defined an analog of the Laplacian on a class of self-similar fractals, including the familiar Sierpinski gasket. We study properties of this operator. We show that there is a maximal principle for solutions of certain nonlinear equations of the form u(x)=F(x, u(x)). We discuss the extension of the Laplacian to non-compact fractal blow-ups, and show that it is essentially self-adjoint, and we prove an analog of Liouville's theorem in some cases. We also give an explicit algorithm for solving the Dirichlet problem for certain domains in the Sierpinski gasket and give a characterization of all harmonic functions on those domains. Copyright 1999 Academic Press.