Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider a domain in Euclidean N-dimensional space whose boundary is the entire graph of a continuous real-valued function on an (N-1)-dimensional subspace. In the domain we consider the heat flow with initial temperature zero and boundary temperature one. The problem is to characterize the boundary of such a kind of domain if this contains a stationary isothermic surface, which is a level surface of temperature which does not evolve with time. We show that the boundary of the domain must be a hyperplane under some general conditions on its boundary. This is related to Liouville or Bernstein-type theorems for certain elliptic Monge-Ampère-type equation.