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Kigami, Jun; Strichartz, Robert S.; Walker, Katharine C.

Constructing a Laplacian on the diamond fractal. (English) Zbl 0999.31007

Exp. Math. 10, No. 3, 437-448 (2001).
A Laplace operator on a finitely ramified (p.c.f. self-similar) fractal can be constructed, for example, by the methods of J. Kigami [“Analysis on fractals”. Cambridge Tracts in Math. 143, Cambridge Univ. Press (2001; Zbl 0998.28004)] provided a nonlinear finite-dimensional eigenvalue problem possesses a unique solution (up to positive multiples). The authors consider the diamond fractal which is only p.c.f. self-similar in a generalized sense because it seems to be partly finitely, partly infinitely ramified. But, interpreted as a graph-directed construction in the sense of R. D. Mauldin and S. C. Williams [Trans. Am. Math. Soc. 309, 811-829 (1988; Zbl 0706.28007)], it turns out be finitely ramified. Thus the eigenvalue equation can be solved algebraically. The uniqueness of the solution (up to positive multiples) remains open but experimental evidence for it is provided.
Reviewer: Volker Metz (Bielefeld)

Cited in 18 Documents

MSC:

31C20 Discrete potential theory
28A80 Fractals

Keywords:

Laplace operator; fractals; nonlinear eigenvalue problem

Citations:

Zbl 0998.28004; Zbl 0706.28007
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References:

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