zbMATH — the first resource for mathematics

A harmonic calculus on the Sierpiński spaces. (English) Zbl 0686.31003
Starting with an equilateral triangle in $$R^ 2$$ and successively removing, ad infinitum, equilateral triangles whose vertices are midpoints of the equilateral triangles of the preceding generation, W. Sierpiński obtained a gasket-like compact metric space, here denoted $$K^ 3\subset R^ 2$$. The present author generalizes this construction starting with a regular (equilateral) simplex in $$R^{N-1}$$ to obtain $$K^ N\subset R^{N-1}$$ of Hausdorff dimension $$\log N/\log 2.$$ This paper is devoted to the construction of a theory of harmonic functions on $$K^ N$$, called harmonic calculus. The topics covered are harmonic differences, harmonic functions and their series expansion. The latter notions are used to study the analogues of Laplace operator, Poisson equation, Dirichlet problem, Neumann derivatives and Gauss-Green formula. The exposition is very painstaking. It is surprising to this reviewer that such results similar to the classical, smooth case are valid in the present context.
Reviewer: E.J.Akutowicz

MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 55M10 Dimension theory in algebraic topology
Full Text:
References:
 [1] M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket. Prob. Theo. Rel. Fields,79 (1988), 543–624. · Zbl 0635.60090 [2] F. M. Dekking, Recurrent sets. Adv. in Math.,44 (1982), 78–104. · Zbl 0495.51017 [3] K. J. Falconer, The Geometry of Fractal Sets. Cambridge, 1985. · Zbl 0587.28004 [4] M. Hata, On the structure of self-similar sets. Japan J. Appl. Math.,2 (1985), 381–414. · Zbl 0608.28003 [5] M. Hata and M. Yamaguti, The Takagi function and its generalization. Japan J. Appl. Math.,1 (1984), 183–199. · Zbl 0604.26004 [6] J. E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J.,30 (1981), 713–747. · Zbl 0598.28011 [7] S. Kusuoka, A diffusion process on a fractal. Probabilistic Methods in Mathematical Physics, Taniguchi Symp., Katata 1985 (eds. K. Ito, N. Ikeda), Kinokuniya-North Holland, 1987, 251–274. [8] B. B. Mandelbrot, The Fractal Geometry of Nature. W. H. Freeman, San Francisco, 1982. · Zbl 0504.28001 [9] P. A. P. Moran, Additive functions of interval and Hausdorff measure. Proc. Camb. Philos. Soc.,42 (1946), 15–23. · Zbl 0063.04088 [10] W. Sierpinski, Sur une courbe dont tout point est un point de ramification. C. R. Acad. Sci. Paris,160 (1915), 302–305. · JFM 45.0628.02 [11] M. Yamaguti and J. Kigami, Some remarks on Dirichlet problem of Poisson equation. Analyse Mathematique et Applications, Gauthier-Villars, Paris, 1988, 465–471. · Zbl 0672.35018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.