TY - JOUR

T1 - Stationary isothermic surfaces in Euclidean 3-space

AU - Magnanini, Rolando

AU - Peralta-Salas, Daniel

AU - Sakaguchi, Shigeru

N1 - Funding Information:
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 Probabilità e le loro Applicazioni (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica (INdAM).
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

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 -