Stationary isothermic surfaces in Euclidean 3-space

Rolando Magnanini, Daniel Peralta-Salas, Shigeru Sakaguchi

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)97-124
Number of pages28
JournalMathematische Annalen
Volume364
Issue number1-2
DOIs
Publication statusPublished - 2016 Feb 1

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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