Spectral asymptotics for variational fractals
type: Item
instance of: scholarly article
title: Spectral asymptotics for variational fractals
zbMATH Open document ID: 0914.47024
DOI: 10.4171/ZAA/830
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-06-23T00:00:00Z
review text: Summary: A generalization of a classic result of H. Weyl concerning the asymptotics of the spectrum of the Laplace operator is proved for variational fractals. Physically we are studying the density of states for the diffusion through a fractal media. A variational fractal is a couple \((K,{\mathcal E})\) where \(K\) is a self-similar fractal and \({\mathcal E}\) is an energy form with some similarity properties connected with those of \(K\). In this class, we can find some of the most widely studied families of fractals as nested fractals, p.c.f. fractals, the Sierpinski carpet etc., as well as some regular self-similar Euclidean domains. We will see that if \(r(x)\) is the number of eigenvalues associated with \({\mathcal E}\) smaller than \(x\), then \(r(x)\sim x^{\nu/2}\), where \(\nu\) is the intrinsic dimension of \((K,{\mathcal E})\).
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 1182356
zbMATH Keywords: Brownian motion
zbMATH Keywords: spectral dimension
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: asymptotics of the spectrum
zbMATH Keywords: Laplace operator
zbMATH Keywords: variational fractals
zbMATH Keywords: nested fractals
zbMATH Keywords: Sierpinski carpet
author: Gustavo Posta
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1182356