Area versus capacity and solidification in the crushed ice model
type: Item
instance of: scholarly article
title: Area versus capacity and solidification in the crushed ice model
zbMATH Open document ID: 1064.60167
publication date: 2005-02-08T00:00:00Z
review text: Let \(K\) be a non-polar compact set from \(\mathbb{R}^ {m}\), \(m\geq 2\); let \(u:(\mathbb{R}^ {m} \setminus K)\times [0, +\infty ) \rightarrow \mathbb{R}\) be the unique weak solution of the heat equation \[ \Delta u = \frac{\partial u}{\partial t},\;x\in \mathbb{R}^ {m} \setminus K,\;t>0; \quad u(x; 0) = 0,\;x\in \mathbb{R}^ {m} \setminus K; \quad u(x; t) = 1,\;x\in \partial K,\;t > 0. \] The authors give some examples where the geometrical conditions considered by \textit{M. van den Berg} and \textit{J.-F. Le Gall} [Math. Z. 215, No. 3, 437--464 (1994; Zbl 0791.58089)] are not satisfied, and the asymptotic behaviour of the total heat flow \(E_ {K}(t):= \int _ {\mathbb{R}^ {m} \setminus K} u(x; t) \,d x\) is different from \(2 \pi ^ {-1/2} A(\partial K) t^ {1/2} + o (t^ {1/2})\) as \(t\rightarrow 0\) (where \(A(\partial K)\) denotes the area of the boundary of \(K\)). In the examples, \(K\) is the closure of the union of some infinite sequence of disjoint, closed balls (``crushed ice model''). The main tool in the proof is the probabilistic interpretation of the weak solution \(u\).
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 2132289
zbMATH Keywords: heat equation
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: total heat flow
author: M. van den Berg
author: Erwin Bolthausen
MaRDI profile type: Publication
full work available at URL: s00440-003-0332-9
OpenAlex ID: W1981491499
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DOI: 10.1007/S00440-003-0332-9
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2132289