### Abstract

Let Ω be a domain in R^{3} with (Formula presented.), where ∂Ω is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂_{t}u=Δ_{u} over R^{3}, where the initial data is the characteristic function of the set Ω^{c}=R^{3}\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 R^{3}, 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.

Original language | English |
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Pages (from-to) | 97-124 |

Number of pages | 28 |

Journal | Mathematische Annalen |

Volume | 364 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2016 Feb 1 |

### Keywords

- 35K15
- 58J70
- Primary 35K05
- Secondary 53A10

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematische Annalen*,

*364*(1-2), 97-124. https://doi.org/10.1007/s00208-015-1212-1