A characterization of heat balls by a mean value property for temperatures (Q2716115)

\textit{Ü. Kuran} [Bull. Lond. Math. Soc. 4, 311-312 (1972; Zbl 0257.31006)] proved that, if \(D\) is an open set in \(\mathbb{R}^n\) with finite volume \(|D|\) such that \(0\in D\), and if \(h(0)=|D|^{-1} \int_D h\) for every integrable harmonic function \(h\) on \(D\), then \(D\) is a ball centred at the origin. The present paper is concerned with obtaining an analogous result for the heat equation. Noting that the boundary of a ball is a level surface of the Newtonian (or logarithmic, if \(n=2\)) kernel, the authors proceed to consider domains bounded by level surfaces of the Gauss-Weierstrass kernel \(W(x,t)\) on \(\mathbb{R}^n\times \mathbb{R}\), given by \((4\pi t)^{-n/2}\exp(-\|x\|^2/4t)\) if \(t> 0\), and by \(0\) if \(t\leq 0\). Thus they give a ``mean value'' characterization of the heat ball \(\{(x, t)\in \mathbb{R}^{n+1}: W(x_0- x, t_0-t)> (4\pi c)^{-n/2}\}\) with ``centre'' \((x_0, t_0)\) and ``radius'' \(c\). The characterization is more complicated than in the case of harmonic functions.