We consider a two-phase heat conductor in RN with N ≥ 2 consisting of a core and a shell with different constant conductivities. Suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then he structure of the conductor must be spherical. Also, when the medium utside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem with N ≥ 3 and the initial condition where the conductor has temperature 0 and the outside medium has emperature 1. Then we show that almost the same proposition holds true.
