A symmetry theorem in two-phase heat conductors

Hyeonbae Kang; Shigeru Sakaguchi; Hyeonbae Kang; Shigeru Sakaguchi

doi:10.3934/mine.2023061

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-7. doi: 10.3934/mine.2023061

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A symmetry theorem in two-phase heat conductors

Hyeonbae Kang ¹,
Shigeru Sakaguchi ^{2
,
,}

1.
Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, S. Korea
2.
Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

Received: 26 October 2022 Revised: 16 November 2022 Accepted: 16 November 2022 Published: 23 November 2022

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 medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $ C^{2, \alpha} $, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.
- heat diffusion equation,
- two-phase heat conductors,
- Cauchy problem,
- stationary isothermic surface,
- method of moving planes,
- transmission conditions
Citation: Hyeonbae Kang, Shigeru Sakaguchi. A symmetry theorem in two-phase heat conductors[J]. Mathematics in Engineering, 2023, 5(3): 1-7. doi: 10.3934/mine.2023061

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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 medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $ C^{2, \alpha} $, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.

References

[1]	A. D. Alexandrov, Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ., 13 (1958), 5–8.
[2]	D. G. Aronson, Bounds for the fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890–896. https://doi.org/10.1090/S0002-9904-1967-11830-5 doi: 10.1090/S0002-9904-1967-11830-5
[3]	L. Cavallina, R. Magnanini, S. Sakaguchi, Two-phase heat conductors with a surface of the constant flow property, J. Geom. Anal., 31 (2021), 312–345. https://doi.org/10.1007/s12220-019-00262-8 doi: 10.1007/s12220-019-00262-8
[4]	L. Cavallina, S. Sakaguchi, S. Udagawa, A characterization of a hyperplane in two-phase heat conductors, Commun. Anal. Geom., in press.
[5]	E. B. Fabes, D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), 327–338. https://doi.org/10.1007/BF00251802 doi: 10.1007/BF00251802
[6]	B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via maximum principle, Commun. Math. Phys., 68 (1979), 209–243. https://doi.org/10.1007/BF01221125 doi: 10.1007/BF01221125
[7]	H. Kang, S. Sakaguchi, Large time behavior of temperature in two-phase heat conductors, J. Differ. Equations, 303 (2021), 268–276. https://doi.org/10.1016/j.jde.2021.09.027 doi: 10.1016/j.jde.2021.09.027
[8]	W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381–394. https://doi.org/10.1007/s002050050034 doi: 10.1007/s002050050034
[9]	S. Sakaguchi, Two-phase heat conductors with a stationary isothermic surface, Rend. Ist. Mat. Univ. Trieste, 48 (2016), 167–187. https://doi.org/10.13137/2464-8728/13155 doi: 10.13137/2464-8728/13155
[10]	S. Sakaguchi, Two-phase heat conductors with a stationary isothermic surface and their related elliptic overdetermined problems, RIMS Kôkyûroku Bessatsu, B80 (2020), 113–132.
[11]	S. Sakaguchi, Some characterizations of parallel hyperplanes in multi-layered heat conductors, J. Math. Pure. Appl., 140 (2020), 185–210. https://doi.org/10.1016/j.matpur.2020.06.007 doi: 10.1016/j.matpur.2020.06.007
[12]	B. Sirakov, Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 135–156. https://doi.org/10.1016/S0294-1449(00)00052-4 doi: 10.1016/S0294-1449(00)00052-4
[13]	J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304–318. https://doi.org/10.1007/BF00250468 doi: 10.1007/BF00250468

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