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We consider a two-phase heat conductor in two dimensions consisting of a coreand a shell with different constant conductivities. When the medium outside thetwo-phase conductor has a possibly different conductivity, we consider theCauchy problem in two dimensions where initially the conductor has temperature0 and the outside medium has temperature 1.  It is shown that, if there is a stationary isothermic surface in the shellnear 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 ofall the previous results of the author in $N(\ge 2)$ dimensions and twotheorems on their related two-phase elliptic overdetermined problems.

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