A counterexample to Payne’s nodal line conjecture with few holes
Joel Dahné, Javier Gómez-Serrano, Kimberly Hou · Communications in Nonlinear Science and Numerical Simulation · 2021
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.