2008/01/27 by Valentin A. Zagrebnov, Valentin Zagrebnov, Zagrebnov, Valentin
Computer Science · Mathematics · #47D03 #81Q10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #PACS: 47A55 #Spectral Theory in Mathematical Physics #math.FA #msc:47A55 #msc:47D03 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.0801.4145
arxiv created 2008/01/27 · openalex publication_date 2008/01/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper gives a short account of some basic properties of Dirichlet-to-Neumann operators Λγ,∂Ω including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (γ≠ I) and by elliptic systems with dynamical boundary conditions. For illustration of these notions and the properties we use the explicitly constructed Lax semigroups. We demonstrate that for a general smooth bounded convex domain Ω⊂ ℝd the corresponding Dirichlet-to-Neumann semigroup \U(t):= e^-t Λγ,∂Ω\t≥0 in the Hilbert space L2(∂ Ω) belongs to the trace-norm von Neumann-Schatten ideal for any t>0. This means that it is in fact an immediate Gibbs semigroup. Recently Emamirad and Laadnani have constructed a Trotter-Kato-Chernoff product-type approximating family \(Vγ, ∂Ω(t/n))n \n ≥ 1 strongly converging to the semigroup U(t) for n→∞. We conclude the paper by discussion of a conjecture about convergence of the Emamirad-Laadnani approximantes in the the trace-norm topology.