Two-phase heat conductors with a stationary isothermic surface

TL;DR

This paper investigates conditions under which a two-phase heat conductor with a stationary isothermic surface must be spherical, revealing geometric constraints based on thermal boundary conditions and conductivities.

Contribution

It establishes that the presence of a stationary isothermic surface implies the conductor's sphericity, extending to cases with different external conductivities and initial conditions.

Findings

Stationary isothermic surface implies spherical shape.

Results hold for different external conductivities.

Extension to higher dimensions and various initial conditions.

Abstract

We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ 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 the structure of the conductor must be spherical. Also, when the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem with $N \ge 3$ and the initial condition where the conductor has temperature 0 and the outside medium has temperature 1. Then we show that almost the same proposition holds true.