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Metric properties of degenerate and fractals media. (English) Zbl 0894.28002
Cioranescu, Doina (ed.) et al., Homogenization and applications to material sciences. Proceedings of the international conference, Nice, France, June 6–10, 1995. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 9, 291-308 (1995).
The author points to the fact that self-similar fractals \(K\) and degenerate elliptic media can be considered from the standpoint of spaces of homogeneous type. A topological space \(S\) with a quasi-distance \(d\) is said to be of homogeneous type provided that the balls \(B_R\) of \(d\) are a basis of neighbourhoods and \(B_R\) contains at most \(c\varepsilon^{-\nu}\) points with mutual distance \(\geq \varepsilon R\) with constants \(\nu>0\) and \(c\) independent of the \(B_R\)’s. Such a metric on the Euclidean space \(\mathbb{R}^D\) arises from the setting for a Hörmander type sub-elliptic operator. The constant \(\nu \) becomes the homogeneous dimension. In the case of self-similar fractals \(K\) Dirichlet energy forms \(E\) must be considered. The quasi-distance \(d\) is definded by \[ d(x,y)=| x-y|^\delta\quad\text{for }x,y\in K, \] where \(|.|\) is the Euclidean distance, but \(\delta\) is determined by requiring that \(d^{2}\) scales as \(E\) on \(K\).
For the entire collection see [Zbl 0873.00028].

28A80 Fractals
35J99 Elliptic equations and elliptic systems