Communications in Analysis and Geometry

Volume 31 (2023)

Number 7

A characterization of a hyperplane in two-phase heat conductors

Pages: 1867 – 1888

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n7.a9

Authors

Lorenzo Cavallina (Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan)

Shigeru Sakaguchi (Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan)

Seiichi Udagawa (Department of Mathematics, School of Medicine, Nihon University, Itabashi, Tokyo 173-0032, Japan)

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 has temperature 0 and the other has temperature 1. Suppose that the interface is connected and uniformly of class $C^{6}$. We show that if the interface has a time-invariant constant temperature, then it must be a hyperplane.

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This research was partially supported by the Grants-in-Aid for Scientific Research (B) (#18H01126 and #17H02847) and JSPS Fellows (#18J11430) of Japan Society for the Promotion of Science.

Received 31 December 2019

Accepted 14 October 2020

Published 26 July 2024