A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian
A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction.