Courant-sharp eigenvalues of a two-dimensional torus
Analysis of PDEs
2015-07-16 v2 Spectral Theory
Abstract
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{\'e}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.
Cite
@article{arxiv.1501.02558,
title = {Courant-sharp eigenvalues of a two-dimensional torus},
author = {Corentin Léna},
journal= {arXiv preprint arXiv:1501.02558},
year = {2015}
}