Stationary isothermic surfaces in Euclidean 3-space

Abstract

Let \(\Omega \) be a domain in \({\mathbb R}^3\) with \(\partial \Omega = \partial ({\mathbb R}^3{\setminus } \overline{\Omega })\), where \(\partial \Omega \) is unbounded and connected, and let \(u\) be the solution of the Cauchy problem for the heat equation \(\partial _t u= \Delta u\) over \({\mathbb R}^3,\) where the initial data is the characteristic function of the set \(\Omega ^c = {\mathbb R}^3{\setminus } \Omega \). We show that, if there exists a stationary isothermic surface \(\Gamma \) of \(u\) with \(\Gamma \cap \partial \Omega = \varnothing \), then both \(\partial \Omega \) and \(\Gamma \) must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that \(\Gamma \cap \partial \Omega =\varnothing \) and \(\partial \Omega \) is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in \({\mathbb R}^3\), a notion that was introduced by Magnanini et al. (Trans Am Math Soc 358:4821–4841, 2006). In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.