Stationary isothermic surfaces in Euclidean 3-space

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^3$ with $\partial \Omega =\partial \left({\mathbb{R}}^3\setminus \overline{\Omega }\right)$, 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 {\it uniformly dense domains } in ${\mathbb{R}}^3$, a notion that was introduced by Magnanini, Prajapat \& Sakaguchi in \cite{MPS2006tams}. 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.

31 pages