**SMS scnews item created by Daniel Daners at Fri 7 Mar 2014 1817**

Type: Seminar

Modified: Fri 7 Mar 2014 1820

Distribution: World

Expiry: 10 Mar 2014

**Calendar1: 10 Mar 2014 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@d110-33-139-16.mas800.nsw.optusnet.com.au (ddan2237) in SMS-WASM

### PDE Seminar

# On Payne’s nodal line conjecture

### Kennedy

James Kennedy

University of Stuttgart, Germany

10th March 2014 14:05-14:55, Carslaw Room 829 (AGR)

## Abstract

A theorem due to Courant states that the zero (nodal) set of any eigenfunction associated
with the $n$th
eigenvalue of the Dirichlet Laplacian on a domain
$\Omega $ divides
$\Omega $ into at
most $n$
connected components, called nodal domains; it is a long-standing area of research
in PDEs and mathematical physics to try to determine how properties of
$\Omega $, such
as its geometry, affect the number and location of these nodal domains.

An old conjecture attributed to Payne asserts that, in the case of the second eigenvalue,
where there are exactly two such nodal domains, both of these should touch the boundary
of $\Omega $;
put differently, the nodal set should not be compactly contained in
$\Omega $.

Along with a short introduction to the conjecture and its history, we
will present two new negative results: firstly, that a simply connected
counterexample can be found in dimension three or higher, and secondly, that
the known counterexample in two dimensions can be adapted to give a
counterexample in the case of Robin boundary conditions. This uses a
new method of proof, which also gives a new proof in the Dirichlet case.

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