**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**

Auth: daners@como.maths.usyd.edu.au

### PDE Seminar

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

### Daners

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.