Closed nodal surfaces for simply connected domains in higher dimensions

Abstract

We give an example of a domain in dimension $N\ge 3$, 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].

10 pages

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