SMS scnews item created by Daniel Daners at Wed 24 Jul 2013 1425
Type: Seminar
Distribution: World
Expiry: 29 Jul 2013
Calendar1: 29 Jul 2013 1400-1500
CalLoc1: AGR Carslaw 829

PDE Seminar

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


Daniel Daners
University of Sydney
Mon 29 July 2013 2-3pm, Carslaw 829 (AGR)


Let \(\Omega\subseteq\mathbb R^N\) be a bounded open set with smooth boundary, and let \(\lambda\in\mathbb R\). The Dirichlet-to-Neumann operator \(D_\lambda\) is a closed operator on \(L^2(\partial\Omega)\) defined as follows. Given \(\varphi\in H^{1/2}(\Omega)\) solve the Dirichlet problem \[ \Delta u+\lambda u=0\quad\text{in \(\Omega\),}\qquad u=\varphi\quad\text{on \(\partial\Omega\).} \] A solution exists if \(\lambda\) is not an eigenvalue of \(-\Delta\) with Dirichlet boundary conditions. If \(u\) is smooth enough we define \[ D_\lambda\varphi:=\frac{\partial u}{\partial\nu}, \] 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 \(e^{-tD_\lambda}\) 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 \(\mathbb R^2\). The example demonstrates some new phenomena: the semigroup \(e^{-tD_\lambda}\) can change from not positive to positive between two eigenvalues. This happens for \(\lambda\in(\lambda_3,\lambda_4)\). Moreover, it is possible that \(e^{-tD_\lambda}\) is positive for large \(t\), but not for small \(t\). The occurrence of such eventually positive semigroups seems to be new. See preprint.

Check also the PDE Seminar page. Enquiries to Florica CÓrstea or Daniel Daners.