2014/07/09 by Rolando Magnanini, Daniel Peralta‐Salas, Magnanini, Rolando +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1407.2419
openalex publication_date 2014/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a domain in \mathbb R3 with ∂Ω= ∂(\mathbb R3∖ Ω), where ∂Ω is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂t u= Δu over \mathbb R3, where the initial data is the characteristic function of the set Ωc = \mathbb R3∖ Ω. We show that, if there exists a stationary isothermic surface Γ of u with Γ∩ ∂Ω= \varnothing, 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=\varnothing and ∂Ω is unbounded. To prove this result, we establish a similar theorem for \it uniformly dense domains in \mathbb R3, a notion that was introduced by Magnanini, Prajapat & Sakaguchi in \citeMPS2006tams. 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.