  We consider the Cauchy problem for the heat diffusion equation in the whole
space consisting of three layers with different constant conductivities, where
initially the upper and middle layers have temperature 0 and the lower layer
has temperature 1. Under some appropriate conditions, it is shown that, if
either the interface between the lower layer and the middle layer is a
stationary isothermic surface or there is a stationary isothermic surface in
the middle layer near the lower layer, then the two interfaces must be parallel
hyperplanes. Similar propositions hold true, either if a stationary isothermic
surface is replaced by a surface with the constant flow property or if the
Cauchy problem is replaced by an appropriate initial-boundary value problem.
