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

TL;DR

This paper constructs a specific domain in the plane where the second Robin eigenfunction's nodal line is closed, providing new insights into eigenfunction behavior and extending known results to Robin boundary conditions.

Contribution

It introduces a novel multiply connected domain with a closed nodal line for the second Robin eigenfunction and adapts a Donnelly-Fefferman type bound for interior nodal set estimates.

Findings

Constructed a domain with a closed interior nodal line for the second Robin eigenfunction.

Established a boundary-independent uniform estimate on the size of nodal sets.

Provided a new proof of the Dirichlet case nodal line property.

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.