Closed nodal surfaces for simply connected domains in higher dimensions

TL;DR

This paper constructs a specific higher-dimensional domain with an analytic boundary where the second eigenfunction of the Dirichlet Laplacian exhibits a closed nodal surface, challenging previous assumptions.

Contribution

It provides the first example of a simply connected, higher-dimensional domain with an analytic boundary having a closed nodal surface for the second eigenfunction.

Findings

Existence of a higher-dimensional domain with a closed nodal surface

Construction via perturbations of a known domain

Counterexample to previous conjectures

Abstract

We give an example of a domain in dimension $N \geq 3$, homeomorphic to a ball and with analytic boundary, for which the second eigenvalue of the Dirichlet Laplacian has an eigenfunction with a closed nodal surface. The domain is constructed via a sequence of perturbations of the domain of S. Fournais [J. Differential Equations \textbf{173} (2001), 145-159].