Stationary isothermic surfaces for unbounded domains

The initial temperature of a heat conductor with non-compact boundary is zero and its boundary temperature is kept equal to one at each time. The conductor contains a stationary isothermic surface, that is, a time-invariant spatial level surface of the temperature. We prove that a kind of Liouville theorem: the unbounded boundary of the conductor, if it satisfies certain sufficient global assumptions, must be either a hyperplane or the union of two parallel hyperplanes. We present two different proofs (based on different assumptions): one is based on a short-time asymptotic formula for the heat content of a ball touching the boundary from inside and on a Bernstein-type theorem; the other one relies on a sliding method due to Berestycky, Caffarelli and Nirenberg.