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, Sendai, Japan)

Shigeru Sakaguchi (Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan; and Institute for Excellence in Higher Education, Tohoku University, Sendai, Japan)

Seiichi Udagawa (Department of Mathematics, School of Medicine, Nihon University, Tokyo, 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 10 August 2024