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

Abstract

The paper gives a short account of some basic properties of \textit{Dirichlet-to-Neumann} operators $Λ_{γ,\partialΩ}$ including the corresponding semigroups motivated by the Laplacian transport in anisotropic media ($γ\neq I$) and by elliptic systems with dynamical boundary conditions. For illustration of these notions and the properties we use the explicitly constructed \textit{Lax semigroups}. We demonstrate that for a general smooth bounded convex domain $Ω\subset \mathbb{R}^d$ the corresponding {Dirichlet-to-Neumann} semigroup $\left\{U(t):= e^{-t Λ_{γ,\partialΩ}}\right\}_{t\geq0}$ in the Hilbert space $L^2(\partial Ω)$ belongs to the \textit{trace-norm} von Neumann-Schatten ideal for any $t>0$. This means that it is in fact an \textit{immediate Gibbs} semigroup. Recently Emamirad and Laadnani have constructed a \textit{Trotter-Kato-Chernoff} product-type approximating family $\left\{(V_{γ, \partialΩ}(t/n))^n \right\}_{n \geq 1}$ \textit{strongly} converging to the semigroup $U(t)$ for $n\to\infty$. We conclude the paper by discussion of a conjecture about convergence of the \textit{Emamirad-Laadnani approximantes} in the the {\textit{trace-norm}} topology.