From Laplacian Transport to Dirichlet - to - Neumann ( Gibbs ) Semigroups

3 The paper gives a short account of some basic properties of Dirichlet-to-Neumann operators Λ γ,∂Ω including the corresponding semigroups motivated by the Lapla-cian transport in anisotropic media (γ = I) and by elliptic systems with dynam-ical 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 ⊂ R d the corresponding Dirichlet-to-Neumann semigroup U(t) := e −tΛ γ,∂Ω t≥0 in the Hilbert space L 2 (∂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 {(V γ,∂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.