TY - JOUR

T1 - Stationary isothermic surfaces in Euclidean 3-space

AU - Magnanini, Rolando

AU - Peralta-Salas, Daniel

AU - Sakaguchi, Shigeru

N1 - Funding Information:
The authors would like to thank Professor Reiko Miyaoka for her interest in their work and some useful discussions. Also, the authors are grateful to the anonymous referees for giving invaluable suggestions to improve the paper. This research was partially supported by the ERC Grant 335079 and the Spanish MINECO grant SEV-2011-0087, and by Grants-in-Aid for Scientific Research (B) (20340031 and 26287020) and for Challenging Exploratory Research (25610024) of Japan Society for the Promotion of Science. The first and third author have also been supported by the Gruppo Nazionale per l?Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica (INdAM).

PY - 2016/2/1

Y1 - 2016/2/1

N2 - Let Ω be a domain in R3 with (Formula presented.), where ∂Ω is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂tu=Δu over R3, where the initial data is the characteristic function of the set Ωc=R3\Ω. We show that, if there exists a stationary isothermic surface Γ of u with Γ∩∂Ω=∅, then both ∂Ω and Γ must be either parallel planes or co-axial circular cylinders. This theorem completes the classification of stationary isothermic surfaces in the case that Γ∩∂Ω=∅ and ∂Ω is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in R3, a notion that was introduced by Magnanini et al. (Trans Am Math Soc 358:4821–4841, 2006). In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

AB - Let Ω be a domain in R3 with (Formula presented.), where ∂Ω is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂tu=Δu over R3, where the initial data is the characteristic function of the set Ωc=R3\Ω. We show that, if there exists a stationary isothermic surface Γ of u with Γ∩∂Ω=∅, then both ∂Ω and Γ must be either parallel planes or co-axial circular cylinders. This theorem completes the classification of stationary isothermic surfaces in the case that Γ∩∂Ω=∅ and ∂Ω is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in R3, a notion that was introduced by Magnanini et al. (Trans Am Math Soc 358:4821–4841, 2006). In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

KW - 35K15

KW - 58J70

KW - Primary 35K05

KW - Secondary 53A10

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U2 - 10.1007/s00208-015-1212-1

DO - 10.1007/s00208-015-1212-1

M3 - Article

AN - SCOPUS:84955277110

VL - 364

SP - 97

EP - 124

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1-2

ER -