Polygonal heat conductors with a stationary hot spot
Let \(\Omega\) be a convex polygon whose incircle touches every side of \(\partial\Omega\). Consider the heat equation under constant (positive) initial datum and zero boundary conditions. The hot spot is the point at which the temperature attains its spatial maximum at each given time. If the hot spot is stationary, the authors derive two necessary geometric conditions on the geometry of \(\Omega\). In special cases these conditions imply symmetries, provided \(\Omega\) is a pentagon or hexagon.
scientific article; zbMATH DE number 5351755
scientific article