On Payne’s nodal line conjecture

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.