**SMS scnews item created by Daniel Daners at Fri 30 Aug 2019 1532**

Type: Seminar

Distribution: World

Expiry: 2 Sep 2019

**Calendar1: 2 Sep 2019 1200-1300**

**CalLoc1: AGR Carslaw 829**

CalTitle1: PDE Seminar: On the eigenvalues of the Robin Laplacian with a complex parameter (Kennedy)

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

### PDE Seminar

# On the eigenvalues of the Robin Laplacian with a complex parameter

### Kennedy

James Kennedy

University of Lisbon, Portugal

Mon 2nd Sep 2019, 12-1pm, Carslaw Room 829 (AGR)

## Abstract

We are interested in the eigenvalues of the Laplacian on
a bounded domain with boundary conditions of the form
$\frac{\partial u}{\partial \nu}+\alpha u=0$, where
$\nu $ is the outer unit normal
to the boundary and $\alpha $
should be considered a parameter on which the eigenvalues depend.

For positive $\alpha $
this operator, and in particular its eigenvalues, interpolate in a strong sense between those of the
Neumann ($\alpha =0$) and
Dirichlet (formally $\alpha =\infty $)
Laplacians. In recent years, however, the case of large negative
$\alpha $ has been
studied intensively, and in particular the asymptotics of the eigenvalues in the singular
limit $\alpha \to -\infty $
is well understood: there is a sequence of eigenvalues which diverges like
$-{\alpha}^{2}$,
independently of the geometry of the domain, while any non-divergent eigenvalues
converge to points in the spectrum of the Dirichlet Laplacian.

Here, after giving a brief overview of what is known for real
$\alpha $, we
will present a number of new results for the corresponding problem when
$\alpha $ is a
(usually large) complex parameter. This is based on ongoing joing work with
Sabine Bögli (Imperial College London) and Robin Lang (University of
Stuttgart).

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