Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions

TL;DR

This paper extends Pleijel's nodal domain theorem to Neumann and Robin eigenfunctions, demonstrating that equality in Courant's theorem occurs only finitely often for these boundary conditions on bounded domains.

Contribution

It proves a Pleijel-type theorem for Neumann and Robin eigenfunctions, confirming a longstanding conjecture for these cases in all dimensions.

Findings

Equality in Courant's theorem occurs finitely often for Neumann eigenfunctions.

The result extends to Robin boundary conditions.

Confirms Pleijel's conjecture beyond Dirichlet cases.

Abstract

In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary conditions.