The nodal line of the second eigenfunction of the Robin Laplacian in ${\mathbb{R}}^2$ can be closed

Abstract

We construct a multiply connected domain in ${\mathbb{R}}^2$ for which the second eigenfunction of the Laplacian with Robin boundary conditions has an interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains, which does not depend directly on any boundary behaviour. This also gives a new proof of the nodal line property of the example in the Dirichlet case.

20 pages

References in corpus (1)

• Closed nodal surfaces for simply connected domains in higher dimensions