Area versus capacity and solidification in the crushed ice model (Q706331)

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 \(u\).