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

# 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.

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