Perimeter on fractal sets
type: Item
instance of: scholarly article
title: Perimeter on fractal sets
zbMATH Open document ID: 0727.28004
DOI: 10.1007/BF02568263
author: Andrea Braides
author: Piero D'ancona
published in: Manuscripta Mathematica
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155626
review text: The authors remark that the Calculus of Variations and Geometric Measure Theory does not seem to have an easy extension to fractal problems. Their aim is to give a contribution to the problem defining lower semicontinuous functionals. They consider the following special case: Let \(V_ 0\) be an equilateral triangle of side length 1 with center in 0, \(V_{h+1}\) the union of \(V_ h\) and \(3\cdot 4^ h\) equilateral triangles of side length \(3^{-(h+1)}\) external to \(V_ h\), such that one of their edges lies on the central third of one of the \(3\cdot 4^ h\) edges of \(V_ h\). Then \(V=\overline{\cup V_ h}\) is the full snowflake whose boundary is composed of three copies of the Koch curve. Let denote \(\partial E\) the topological boundary of the set E and \(\partial^*E\) its essential boundary, that is the set \[ \partial^*E=\{x\in {\mathbb{R}}^ 2:\limsup_{\rho \to 0^+}\frac{| B_{\rho}(x)\setminus E|}{\rho^ n}>0,\quad \limsup_{\rho \to 0^+}\frac{| B_{\rho}\cap R|}{\rho^ n}>0\}. \] If A is any open set and E any set of finite perimeter the authors study the behaviour of the sequence of functionals \[ F_ h(E,A)={\mathcal H}_ 1(A\cap (\partial^*E\setminus \partial^*V_ h))+(3/4)^ h{\mathcal H}_{\alpha}(K){\mathcal H}_ 1(A\cap \partial^*E\cap \partial^*V_ h), \] where \({\mathcal H}_ 1,{\mathcal H}_{\alpha}\) are the 1-respectively \(\alpha\)-dimensional Hausdorff measure \((\alpha =\log 4/\log 3)\) and K stands for the Koch curve. Provided that E is measurable and \((\partial E\cap A)\setminus \partial^*V\) consists of a finite number of regular arcs then the so called \(\Gamma\)-limit of \((F_ n(E,A))\) in the sense of De Giorgi exists and is equal to \[ {\mathcal H}_ 1(A\cap (\partial^*E\setminus \partial^*V))+{\mathcal H}_{\alpha}(A\cap \partial^*E\cap \partial^*V) \] and thus becomes a lower semicontinuous functional.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 4200524
zbMATH Keywords: fractal set
zbMATH Keywords: snowflake
zbMATH Keywords: Koch curve
zbMATH Keywords: finite perimeter
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: \(\Gamma \) -limit
zbMATH Keywords: lower semicontinuous functional
MaRDI profile type: Publication
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OpenAlex ID: W2052058783
description: scientific article; zbMATH DE number 4200524