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Variational fractals. (English) Zbl 1016.28010
A variational fractal is a triple \((K,\mu,E)\), where \(K\) is a self-similar set in the Euclidean space satisfying the open-set condition and a certain “boundary” condition, \(\mu\) is the normalized \(d_f\)-dimensional Hausdorff measure restricted to \(K\), where \(d_f\) denotes the Hausdorff dimension of \(K\), and \(E\) is a symmetric Dirichlet form on \(L^2(K,\mu)\) satisfying five somewhat technical conditions. Examples of variational fractals include nested fractals with the “standard” form \(E\) constructed by T. Lindstrøm [Mem. Am. Math. Soc. 420 (1990; Zbl 0688.60065)] and M. Fukushima [in: Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), 151-161 (1992; Zbl 0764.60081)]. For each \(\delta> 0\), the author equips \(K\) with the quasi-metric \(d(x,y)=|x-y|^\delta\) for \(x,y\in K\), and studies various Poincaré-type inequalities associated with \(d\) and the Dirichlet form \(E\). The main result in the paper states that if \(\delta\) equals the intrinsic dimension of \(K\), then the following scaled Poincaré inequality holds: There exist \(q\geq 1\) and \(C> 0\) such that \[ \int_{B(x,r)}|u-\overline u_{B(x,r)}|^2\mu(dy)\leq C\Biggl({r\over\text{diam }K}\Biggr)^2 \int_{B(x,qr)}\gamma[u] (dy) \] for \(u\) in the domain of \(E\), \(x\in K\) and \(0< r<\text{diam }K\), where \(B(x,r)\) denotes the ball in \(K\) with respect to the quasi-metric \(d\), \(\gamma[u]\) denotes the so-called local energy measure associated with \(E(u,u)\), and \(\overline u_{B(x,r)}= \int_{B(x,r)}\overline u\mu(dy)\), where \(\overline u\) denotes the quasi-continuous representative of \(u\). Finally, the author considers some examples.

MSC:
28A80 Fractals
26D10 Inequalities involving derivatives and differential and integral operators
31C25 Dirichlet forms
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J45 Methods involving semicontinuity and convergence; relaxation
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