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

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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.

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Additional Information

**Robert S. Strichartz**

Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Email:
str@math.cornell.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03685-2

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