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

TL;DR

This paper investigates the conditions under which a two-phase heat conductor with a stationary isothermic surface must be circular, linking geometric properties to elliptic overdetermined problems in two dimensions.

Contribution

It establishes that the presence of a stationary isothermic surface implies the conductor's structure is circular, extending previous results to two dimensions and related elliptic problems.

Findings

Stationary isothermic surface implies circular structure

Provides new proofs for higher-dimensional cases

Introduces two theorems on elliptic overdetermined problems

Abstract

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.