# Some characterizations of parallel hyperplanes in multi-layered heat conductors

## TL;DR

This paper investigates conditions under which interfaces in a multi-layered heat conductor must be parallel hyperplanes, based on stationary isothermic surfaces and flow properties, revealing geometric constraints in heat diffusion problems.

## Contribution

It characterizes when interfaces in layered heat conductors are necessarily parallel hyperplanes based on stationary isothermic and flow conditions, extending previous geometric results.

## Key findings

- Interfaces are parallel hyperplanes under certain stationary isothermic conditions.
- Similar results hold for surfaces with constant flow properties.
- Findings apply to both Cauchy and initial-boundary value problems.

## Abstract

We consider the Cauchy problem for the heat diffusion equation in the whole space consisting of three layers with different constant conductivities, where initially the upper and middle layers have temperature 0 and the lower layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the lower layer and the middle layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the lower layer, then the two interfaces must be parallel hyperplanes. Similar propositions hold true, either if a stationary isothermic surface is replaced by a surface with the constant flow property or if the Cauchy problem is replaced by an appropriate initial-boundary value problem.