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

Abstract

Let $\Omega \subseteq {ℝ}^N$ be a bounded open set with smooth boundary, and let $\lambda \in ℝ$. The Dirichlet-to-Neumann operator $D_{\lambda }$ is a closed operator on $L^2\left(\partial \Omega \right)$ defined as follows. Given $\phi \in H^{1∕2}\left(\Omega \right)$ solve the Dirichlet problem

A solution exists if $\lambda$ is not an eigenvalue of $-\Delta$ with Dirichlet boundary conditions. If $u$ is smooth enough we define

where $\nu$ is the outer unit normal to $\partial \Omega$. Let $0<{\lambda }_1<{\lambda }_2<{\lambda }_3<\dots$ be the strictly ordered Dirichlet eigenvalues of $-\Delta$ on $\Omega$. It was shown by Arendt and Mazzeo that $e^{-t{D}_{\lambda }}$ is positive and irreducible if $\lambda <{\lambda }_1$. The question left open was whether or not the semigroup is positive for any $\lambda >{\lambda }_1$. The aim of this talk is to explore this question by explicitly computing the semigroup for the disc in ${ℝ}^2$. The example demonstrates some new phenomena: the semigroup $e^{-t{D}_{\lambda }}$ can change from not positive to positive between two eigenvalues. This happens for $\lambda \in \left({\lambda }_3,{\lambda }_4\right)$. Moreover, it is possible that $e^{-t{D}_{\lambda }}$ is positive for large $t$, but not for small $t$. The occurrence of such eventually positive semigroups seems to be new.

A preprint is available.