From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups - Zagrebnov, V. A. (Журнал математичної фізики, аналізу, геометрії, 2008)
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 Laplacian transport in anisotropic media (γ eq I) and by elliptic systems with dynamical boundary conditions. To illustrate these notions and the properties we use the explicitly constructed Lax semigroups. We demonstrate that for a general smooth bounded convex domain ΩR^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 H. Emamirad and I. 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 trace-norm topology. Mathematics Subject Classification: 47A55, 47D03, 81Q10.