Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analysis on products of fractals

Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 357 (2005), 571-615
MSC (2000): Primary 31C45, 28A80
Published electronically: September 23, 2004
MathSciNet review: 2095624
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a class of post-critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non-p.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the non-p.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 31C45, 28A80

Retrieve articles in all journals with MSC (2000): 31C45, 28A80

Additional Information

Robert S. Strichartz
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Received by editor(s): July 8, 2003
Published electronically: September 23, 2004
Additional Notes: The author’s research was supported in part by the National Science Foundation, grant DMS–0140194
Article copyright: © Copyright 2004 American Mathematical Society