Two-phase heat conductors with a stationary isothermic surface and their related elliptic overdetermined problems

We consider a two-phase heat conductor in two dimensions consisting of a core and a shell with different constant conductivities. When the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem in two dimensions where initially the conductor has temperature 0 and the outside medium has temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be circular. Moreover, as by-products of the method of the proof, we mention other proofs of all the previous results of the author in $N(\ge 2)$ dimensions and two theorems on their related two-phase elliptic overdetermined problems.