A counterexample to Payne's nodal line conjecture with few holes

Abstract

Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with many holes and raised the question of the minimum number of holes a counterexample can have. In this paper we prove it is at most 6.