A symmetry theorem in two-phase heat conductors

TL;DR

This paper proves that in a two-phase heat conductor with stationary isothermic interface, the interface must be spherical, using the method of moving planes, under certain regularity and boundedness conditions.

Contribution

It establishes a symmetry result for stationary isothermic interfaces in two-phase heat conductors, extending the understanding of shape properties in heat diffusion problems.

Findings

Stationary isothermic interface implies spherical shape.

Method of moving planes is effective for this symmetry proof.

Results apply to bounded media with smooth interfaces.

Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $C^{2,\alpha}$, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.