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Zbl 0686.31003
Kigami, Jun
A harmonic calculus on the Sierpiński spaces. (English)
[J] Japan J. Appl. Math. 6, No.2, 259-290 (1989). ISSN 0910-2043

Starting with an equilateral triangle in $R\sp 2$ 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 $K\sp 3\subset R\sp 2$. The present author generalizes this construction starting with a regular (equilateral) simplex in $R\sp{N-1}$ to obtain $K\sp N\subset R\sp{N-1}$ of Hausdorff dimension $\log N/\log 2.$ This paper is devoted to the construction of a theory of harmonic functions on $K\sp N$, 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.
[E.J.Akutowicz]

MSC 2000:: *31B20 Boundary value and inverse problems (higher-dim. potential theory)
35J05 Laplace equation, etc.
55M10 Dimension theory (algebraic topology)

Keywords: Sierpiński spaces; Dirichlet problem; Neumann derivatives; Gauss-Green formula; harmonic differences; series expansion

Cited in: Zbl 1082.31004 Zbl 0735.42018

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