Area versus capacity and independence in the crushed ice model
type: Item
instance of: scholarly article
title: Area versus capacity and independence in the crushed ice model
zbMATH Open document ID: 1010.31004
published in: Journal of Functional Analysis
publication date: 2003-05-18T00:00:00Z
review text: The paper studies the heat flow \(E_K(t)= \int_{K^c} u(x,t) dt\) from the closure \(K\) of an infinite union \(B\) of disjoint closed balls having radii \(r_1\geq r_2\geq\dots>0\) in \(\mathbb R^m\), \(m\geq 2\), kept at constant temperature \(1\), while the surrounding medium is at temperature \(0\) at time \(0\) (\(u(x,t)\) being the temperature in \(x\) at time \(t\)). The main result is the following: Assume that \(K\setminus B\) has zero capacity, \(\sum_i r_i^{m-2}<\infty\) if \(m\geq 3\), and \(\sum_i \left(\log\frac{2r_1}{r_i}\right)^{-1} <\infty\) if \(m=2\). Then \[ E_K(t)=2\pi^{-1/2} A(\partial K) t^{1/2}+o(t^{1/2})\quad\text{ as }t\to 0 \] where \(A(\partial K)\;(=A(\partial B)!)\) denotes the area of the boundary \(\partial K\) of \(K\). The same behavior of \(E_K(t)\) had been shown by the author and \textit{J. F. Le Gall} [Math. Z. 215, No. 3, 437-464 (1994; Zbl 0791.58089)] for a cooling obstacle \(K\) with smooth boundary.
reviewed by: Wolfhard Hansen
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1747021
zbMATH Keywords: heat equation
zbMATH Keywords: Brownian motion
zbMATH Keywords: probabilistic potential theory
zbMATH Keywords: diffusion processes
author: M. van den Berg
MaRDI profile type: Publication
full work available at URL: jfan.2001.3845
OpenAlex ID: W2129866540
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DOI: 10.1006/JFAN.2001.3845
description: scientific article; zbMATH DE number 1747021