Mathematics – Functional Analysis
Scientific paper
2008-01-27
Mathematics
Functional Analysis
Scientific paper
The paper gives a short account of some basic properties of \textit{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. 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 $\Omega \subset \mathbb{R}^d$ the corresponding {Dirichlet-to-Neumann} semigroup $\left\{U(t):= e^{-t \Lambda_{\gamma,\partial\Omega}}\right\}_{t\geq0}$ in the Hilbert space $L^2(\partial \Omega)$ 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 $\left\{(V_{\gamma, \partial\Omega}(t/n))^n \right\}_{n \geq 1}$ \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.
No associations
LandOfFree
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-313729