Matzoh ball soup: Heat conductors with a stationary isothermic surface (Q1860984)

The main result of the paper is the following theorem: Let \(\Omega \) be a bounded domain in \(\mathbb{R}^N\), \(N\geq 2\), satisfying the exterior sphere condition and suppose that \(D\) is a domain, with boundary \(\partial D\), satisfying the interior cone condition, and such that \(\overline D\subset \Omega \). Let \(u\) be a solution of the problem \(u_t=\Delta u\) in \(\Omega \times (0,+\infty)\), \(u=1\) on \(\partial \Omega \times (0,+\infty)\), \(u=0\) on \(\Omega \times \{ 0\} \). Assume that \(u(x,t)=a(t)\) for \((x,t) \in \partial D \times (0,+\infty)\) for some function \(a:(0,+\infty) \to (0,+\infty)\). Then \(\Omega \) must be a ball.