Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator. (English) Zbl 1375.35139
Summary: 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 \(\varDelta +\lambda I\) as \(\lambda \) varies. It is known that the semigroup is positive if \(\lambda <\lambda _1\), where \(\lambda _1\) is the principal eigenvalue of \(-\varDelta \) 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 \(\lambda >\lambda _1\).
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
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