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

Abstract

By analysing some explicit examples we investigate the positivity and non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator \(\Delta +\lambda I\) as \(\lambda\) varies. It is known that the semigroup is positive if \(\lambda<\lambda_1\), where \(\lambda_1\) is the principal eigenvalue of \(-\Delta\) 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 \(\lambda>\lambda_1\).