2024/08/28 by Shigeru Sakaguchi, Sakaguchi, Shigeru · 1 citation
Engineering · #35B06 #35B40 #35K15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Heat Transfer and Boiling Studies #Heat Transfer and Optimization #Phase Equilibria and Thermodynamics #Primary 35K05 #Secondary 35K10
paper · pdf · doi:10.48550/arxiv.2408.15539
openalex publication_date 2024/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C2 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 C2 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.