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)

Abstract

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 Cîrstea or Daniel Daners.