\(p\)-energy and \(p\)-harmonic functions on Sierpinski gasket type fractals
type: Item
instance of: scholarly article
title: \(p\)-energy and \(p\)-harmonic functions on Sierpinski gasket type fractals
zbMATH Open document ID: 1044.31006
DOI: 10.1023/A:1026377524793
author: Roberto Peirone
author: Robert S. Strichartz
author: Paul Edward Herman
published in: Potential Analysis
publication date: 2004-03-15T00:00:00Z
review text: The classical \(p\)-energy of a function \(u\) is the integral of \(| \nabla u| ^p\), for the gradient \(\nabla\) and \(1< p<\infty\). The techniques of \textit{J. Kigami} [Analysis on fractals. (Cambridge Tracts in Math. 143, Cambridge University Press, Cambridge) (2001; Zbl 0998.28004)] for the case \(p=2\) are extended. The fractal is approximated by a sequence of graphs endowed with their discrete \(p\)-energies. A nonlinear renormalization problem is solved to find an appropriate scaling factor and a corresponding self-similar discrete \(p\)-energy. The construction of the limiting model mimics the case \(p=2\). The existence result relies on Schauder's fixed point theorem but the uniqueness remains open. A function is termed \(p\)-harmonic when it minimizes the \(p\)-energy subject to boundary conditions. Its uniqueness is again only conjectured. Numerical approximations for the Sierpinski gasket are presented. The existence proof is generalized to Peirone's ``weakly completely symmetric fractals'' [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 3, No. 2, 431--460 (2000; Zbl 0958.31005)].
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2057479
zbMATH Keywords: \(p\)-Laplace operator
zbMATH Keywords: fractals
zbMATH Keywords: \(p\)-energy
zbMATH Keywords: sequence of graphs
zbMATH Keywords: \(p\)-harmonic
zbMATH Keywords: Sierpinski gasket
reviewed by: Volker Metz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2057479