Area versus capacity and independence in the crushed ice model
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.
scientific article; zbMATH DE number 1747021