A toy Neumann analogue of the nodal line conjecture

TL;DR

This paper explores an analogue of the nodal line conjecture for Neumann eigenfunctions on symmetric domains, providing both supporting and counterexample results to the proposed symmetry behavior.

Contribution

It introduces a Neumann analogue of Payne's conjecture, proving it for simply connected domains and presenting a counterexample with holes.

Findings

The conjecture holds for all simply connected planar domains.

Counterexample exists for a domain with two holes.

The problem's structure allows stronger results than the original conjecture.

Abstract

We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain $\Omega$ with rotational symmetry of order two (i.e., $x\in\Omega$ iff $-x\in\Omega$) "should normally" be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.