TY - JOUR

T1 - Two-Phase Heat Conductors with a Surface of the Constant Flow Property

AU - Cavallina, Lorenzo

AU - Magnanini, Rolando

AU - Sakaguchi, Shigeru

N1 - Funding Information:
The first and third authors were partially supported by the Grants-in-Aid for Scientific Research (B) ( 26287020, 18H01126), Challenging Exploratory Research ( 16K13768), and JSPS Fellows ( 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.

PY - 2021/1

Y1 - 2021/1

N2 - We consider a two-phase heat conductor in RN with N≥ 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.

AB - We consider a two-phase heat conductor in RN with N≥ 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.

KW - Cauchy problem

KW - Constant flow property

KW - Diffusion equation

KW - Heat equation

KW - Initial-boundary value problem

KW - Overdetermined problem

KW - Symmetry

KW - Transmission condition

KW - Two-phase heat conductor

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U2 - 10.1007/s12220-019-00262-8

DO - 10.1007/s12220-019-00262-8

M3 - Article

AN - SCOPUS:85072024974

VL - 31

SP - 312

EP - 345

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 1

ER -