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Harmonic calculus on P.C.F. self-similar sets. (English) Zbl 0773.31009
The 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.

MSC:
31C05Generalizations of harmonic (subharmonic, superharmonic) functions
31C20Discrete potential theory and numerical methods
31C25Dirichlet spaces
39A10Additive difference equations
39A12Discrete version of topics in analysis
60J99Markov processes
65Z05Applications of numerical analysis to physics