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 $\Lambda_{\gamma,\partial \Omega}$ including the corresponding semigroups motivated by the Laplacian transport in anisotropic media $(\gamma\neq I)$ 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 $\Omega\subset \mathbb{R}^d$ the corresponding Dirichlet-to-Neumann semigroup $\{U(t):=e^{-t\Lambda_{\gamma,\partial \Omega}}\}_{t\geq 0}$ in the Hilbert space $L_2(\partial \Omega)$ 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 $\{(V_{\gamma,\partial \Omega}(t/n))^n\}_{n\geq 1}$ strongly converging to the semigroup $U(t)$ for $n\to \infty$. 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.