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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.15539 |
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Table of Contents:
- 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 $Γ$ of the interface is of class $C^2$ and is stationary isothermic, then the mean curvature of $Γ$ 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.