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 .
Positivity – Springer Journals
Published: May 26, 2013
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