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

Автор(и)

  • V. A. Zagrebnov Centre de Physique Théorique - UMR 6207, Université de la Méditerranée, Luminy - Case 907, 13288 Marseille, Cedex 09, France

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.

Ключові слова:

Laplacian transport, Dirichlet-to-Neumann operators, Dirichlet-to-Neumann semigroups, Gibbs semigroups

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Як цитувати

(1)
Zagrebnov, V. A. From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups. Журн. мат. фіз. анал. геом. 2008, 4, 551-568.

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