标题： 从拉普拉斯传输到狄利克雷-诺伊曼（吉布斯）半群 显示英文标题

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

摘要： 论文简要介绍了某些基本性质的\textit{狄利克雷-诺伊曼}算子$\Lambda_{\gamma,\partial\Omega}$，包括由各向异性介质中的拉普拉斯传输（$\gamma \neq I$）和具有动态边界条件的椭圆系统所激发的相应半群。为了说明这些概念和性质，我们使用了显式构造的\textit{Lax半群}。 我们证明对于一般的光滑有界凸域$\Omega \subset \mathbb{R}^d$，相应的{狄利克雷-诺伊曼}半群$\left\{U(t):= e^{-t \Lambda_{\gamma,\partial\Omega}}\right\}_{t\geq0}$在希尔伯特空间$L^2(\partial \Omega)$中属于\textit{范数迹}伏纳-施瓦茨理想，对于任何$t>0$都成立。这意味着它实际上是一个\textit{立即吉布斯}半群。 最近，Emamirad 和 Laadnani 构造了一个\textit{特里特-卡托-切诺夫}乘积类型逼近族$\left\{(V_{\gamma, \partial\Omega}(t/n))^n \right\}_{n \geq 1}$\textit{强烈地}收敛到半群$U(t)$对于$n\to\infty$。我们通过讨论关于\textit{埃马米拉德-拉德纳尼近似}在{\textit{迹范数}}拓扑中收敛性的猜想来结束本文。 显示英文摘要

摘要： 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.