Two-Phase Heat Conductors with a Surface of the Constant Flow Property

Abstract

We consider a two-phase heat conductor in \({\mathbb {R}}^N\) with \(N \ge 2\) consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property. Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature 0 and outside temperature 1. We then show that a quite similar symmetry result holds true.

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Acknowledgements

The first and third authors were partially supported by the Grants-in-Aid for Scientific Research (B) (\(\sharp \) 26287020, \(\sharp \) 18H01126), Challenging Exploratory Research (\(\sharp \) 16K13768), and JSPS Fellows (\(\sharp \) 18J11430) of Japan Society for the Promotion of Science. The second author was partially supported by an iFUND-Azione 2-2016 grant of the Università di Firenze. The authors are grateful to the anonymous reviewers for their many valuable comments and remarks to improve clarity in many points.

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Correspondence to Shigeru Sakaguchi.

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Cavallina, L., Magnanini, R. & Sakaguchi, S. Two-Phase Heat Conductors with a Surface of the Constant Flow Property. J Geom Anal 31, 312–345 (2021). https://doi.org/10.1007/s12220-019-00262-8

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Keywords

  • Heat equation
  • Diffusion equation
  • Two-phase heat conductor
  • Transmission condition
  • Initial-boundary value problem
  • Cauchy problem
  • Constant flow property
  • Overdetermined problem
  • Symmetry

Mathematics Subject Classification

  • Primary 35K05
  • Secondary 35K10
  • 35B06
  • 35B40
  • 35K15
  • 35K20
  • 35J05
  • 35J25