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

Анотація

The paper gives a short account of some basic properties of Dirichlet- to-Neumann operators Λ _g,∂W including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (g≠ 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 W Ì ℝ^d the corresponding Dirichlet-to-Neumann semigroup {U(t) := e^{-tΛ _g,∂W}}_t≥0 in the Hilbert space L² (∂W) 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_g,∂W(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.