Interaction between initial behavior of temperature and the mean curvature of the interface in two-phase heat conductors

TL;DR

This paper investigates how the initial temperature behavior relates to the interface's mean curvature in two-phase heat conductors, revealing local geometric insights and conditions for interface symmetry under heat diffusion.

Contribution

It introduces a local method to extract the interface's mean curvature from initial temperature behavior and establishes new symmetry results with relaxed regularity assumptions.

Findings

Mean curvature can be derived from initial temperature behavior at the interface.

Stationary isothermic interfaces with $C^2$ regularity have constant mean curvature.

New symmetry theorems for two-phase heat conductors with weaker regularity conditions.

Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media locally with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumption that a part of the interface between two media with different constant conductivities is of class $C^2$ in a neighborhood of a point $x$ on it, we extract the mean curvature of the interface at $x$ from the initial behavior of temperature at $x$. This result is purely local in space. As a corollary, when the whole Euclidean space consists of two media globally with different constant conductivities, it is shown that if a connected component $\Gamma$ of the interface is of class $C^2$ and is stationary isothermic, then the mean curvature of $\Gamma$ must be constant. Moreover, we apply this result to some overdetermined problems for two-phase heat conductors and obtain some symmetry theorems which relax considerably the regularity assumptions of some previous results.