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

TL;DR

This paper constructs a counterexample to Payne's conjecture with at most six holes, showing the second Dirichlet eigenfunction's nodal line need not touch the boundary.

Contribution

It proves that the minimal number of holes in a counterexample to Payne's conjecture is at most six, refining previous understanding.

Findings

Counterexample with up to six holes

Disproof of Payne's conjecture in certain domains

Limits the minimal holes needed for counterexamples

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.