Dirichlet case on simply-connected planar domains remains open

Determine whether the nodal line of any second Dirichlet eigenfunction of the Laplacian on a bounded simply-connected planar domain can be closed, or prove that such a closed nodal line cannot occur in the simply-connected Dirichlet setting.

Background

The paper contrasts the Neumann and Dirichlet settings for nodal lines of second eigenfunctions. While the authors construct Neumann counterexamples on doubly-connected domains, they note that the analogous question in the Dirichlet case for simply-connected planar domains is not settled.

Historically, Payne conjectured that the second Dirichlet eigenfunction cannot have a closed nodal line for any domain, but counterexamples show the statement is false in general. The simply-connected case, however, is still unresolved.

References

All of this is in sharp contrast with what happened for the Dirichlet problem, for which not only is the result far from being straightforward even in the simply-connected case where it remains open but, following a conjecture formulated by Payne in 1967 [Conjecture 5] stating that the second Dirichlet eigenfunction “cannot have a closed nodal line for any domain”, there appeared a string of partial results [Payne2,lin,putter,Melas,d00, fk08,kiwan,mukherjee-saha] and counterexamples [H2ON,fournais01,freitas-krejcirik07,kennedy,dgh21].

— Neumann's nodal line may be closed on doubly-connected planar domains  (2604.03169 - Freitas et al., 3 Apr 2026) in Introduction, paragraph beginning “All of this is in sharp contrast with what happened for the Dirichlet problem”