Stationary hot spots in a grounded heat conductor

Problem's Description

Let $\Omega$ be a bounded convex domain in Euclidean space. Consider in $\Omega$ the solution $u$ of the heat equation, which is initially equal to $1$ in $\Omega$ and is equal to $0$ on the boundary of $\Omega$ for all times. We know that $u$ has only one critical point in $\Omega$ at each time. This point is called the hot spot and in general moves in time. Let $0\in\Omega$; it is known that, if $\Omega$ is invariant under the action of an essential group $G$ of orthogonal transformations with center at $0$, the hot spot does not move and stays at $0$. We say that $G$ is essential if, for any $x\ne 0$, there is $g\in G$ such that $g\,x\ne x$. Chamberland and Siegel conjectured in 1997 that the converse is true: if the hot spot does not move with time, then $\Omega$ is invariant under the action of $G$. The conjecture is true if $\Omega$ is a triangle or a quadrangle. It is also true for pentagons and hexagons whose sides have the same distance from $0$.

1. Article On heat conductors with a stationary hot spot

   • R. Magnanini,
   • S. Sakaguchi

   Annali di Matematica Pura ed Applicata 183, 1-23, 2004

2. Article Polygonal heat conductors with a stationary hot spot

   • R. Magnanini,
   • S. Sakaguchi

   Journal d'Analyse Mathématique 105, 1-18, 2008

3. Article Convex domains with stationary hot spots

   • M. Chamberland,
   • D. Siegel

   Mathematical Methods in the Applied Sciences 20, 1163-1169, 1997