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


  • V.A. Zagrebnov Centre de Physique Theorique - UMR 6207, Universite de la Mediterranee Luminy - Case 907, 13288 Marseille, Cedex 09, France

Ключові слова:

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


The paper gives a short account of some basic properties of Dirichlet- to-Neumann operators Λ g,∂W including the corresponding semigroups motivated by the Laplacian transport in anisotropic media (g≠ 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 W Ì ℝd the corresponding Dirichlet-to-Neumann semigroup {U(t) := e-tΛ g,∂W}t≥0 in the Hilbert space L2 (∂W) 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 {(Vg,∂W(t/n))n}n≥1 strongly converging to the semigroup U(t) for n→∞. 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.




Як цитувати

Zagrebnov, V. From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups. J. Math. Phys. Anal. Geom. 2008, 4, 551-568.





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