# Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator By analysing some explicit examples we investigate the positivity and the non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator $$\varDelta +\lambda I$$ Δ + λ I as $$\lambda$$ λ varies. It is known that the semigroup is positive if $$\lambda <\lambda _1$$ λ < λ 1 , where $$\lambda _1$$ λ 1 is the principal eigenvalue of $$-\varDelta$$ − Δ with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on $$\lambda >\lambda _1$$ λ > λ 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

, Volume 18 (2) – May 26, 2013
22 pages

/lp/springer_journal/non-positivity-of-the-semigroup-generated-by-the-dirichlet-to-neumann-j8YP3ph1mp
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0243-7
Publisher site
See Article on Publisher Site

### Abstract

By analysing some explicit examples we investigate the positivity and the non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator $$\varDelta +\lambda I$$ Δ + λ I as $$\lambda$$ λ varies. It is known that the semigroup is positive if $$\lambda <\lambda _1$$ λ < λ 1 , where $$\lambda _1$$ λ 1 is the principal eigenvalue of $$-\varDelta$$ − Δ with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on $$\lambda >\lambda _1$$ λ > λ 1 .

### Journal

PositivitySpringer Journals

Published: May 26, 2013

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