Minimal number of holes needed in the Robin case is unknown

Determine the minimal number of boundary components required for a bounded planar domain with Robin boundary conditions to admit a second eigenfunction whose nodal line is closed and does not touch the boundary.

Background

The authors note that closed nodal lines for second eigenfunctions are known to occur in the Robin case on multiply-connected planar domains. However, it is not known how many boundary components are minimally necessary to produce such a phenomenon.

This question parallels the resolved existence in multiply-connected domains and seeks a quantitative topological threshold for the Robin problem.

References

In the Robin case, the nodal line may be closed on multiply-connected domains according to [kennedy11], even if the minimal number of holes needed to construct such a domain remains unknown.

— Neumann's nodal line may be closed on doubly-connected planar domains  (2604.03169 - Freitas et al., 3 Apr 2026) in Introduction, paragraph preceding the Topological Nodal Line Conjecture