Harmonic calculus on P.C.F. self-similar sets.

*(English)*Zbl 0773.31009The author defines and studies a class of fractals called “post critically finite, self-similar sets”, such as the Sierpiński gasket. These sets possess a sufficient degree of regularity and symmetry so as to allow formation of manageable difference operators such as a discrete Laplacian. The first part expounds the definition and properties of the basic fractals, the generation of certain difference operators and the notion of quasi-harmonic and harmonic functions as kernels of the relevant difference operators. The latter part deals with analogies to classical potential theory in Euclidean domains: the Dirichlet problem for the Poisson equation, the Gauss-Green formula, Dirichlet forms. It should be mentioned that the present approach is quite different from probabilistic methods that have been applied by Sh. Kusuoka [Probabilistic methods in mathematical physics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1985, 251-274 (1987; Zbl 0645.60081)] and M. T. Barlow and E. A. Perkins [Probab. Theory Relat. Fields 79, No. 4, 543-623 (1988; Zbl 0635.60090)]. The heavy notations make the paper difficult to read.

Reviewer: E.J.Akutowicz (Montpellier)

##### MSC:

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

31C20 | Discrete potential theory |

31C25 | Dirichlet forms |

39A10 | Additive difference equations |

39A12 | Discrete version of topics in analysis |

60J99 | Markov processes |

65Z05 | Applications to the sciences |