Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions
Analysis of PDEs
2016-12-15 v2 Spectral Theory
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.
Keywords
Cite
@article{arxiv.1609.02331,
title = {Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions},
author = {Corentin Léna},
journal= {arXiv preprint arXiv:1609.02331},
year = {2016}
}
Comments
New version: additional section treating a class of Robin boundary conditions; revision of Section 2.5; additional minor corrections