Topological Nodal Line Conjecture (Dirichlet/Robin)

Establish the Topological Nodal Line Conjecture asserting that, for the Laplacian on a bounded regular planar domain with either Dirichlet or Robin boundary conditions, the nodal line of any second eigenfunction is not closed when the domain is simply-connected, while there exist doubly-connected domains for which the nodal line is closed and does not touch the boundary.

Background

Motivated by parallels across boundary conditions (Dirichlet, Neumann, Robin, Steklov), the authors propose a unifying topological principle: simple connectivity forbids a closed nodal line for the second eigenfunction, whereas doubly-connected domains can exhibit closed nodal lines.

Their work proves the Neumann counterpart on doubly-connected domains and references known results and conjectures in the Dirichlet and Robin cases, prompting a general conjecture encompassing Dirichlet and Robin boundary conditions.