Geodesic
On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I
Documenta mathematica, Tome 23 (2018), pp. 1561-1585.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

According to Courant's theorem, an eigenfunction associated with the $n$-th eigenvalue $\lambda_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $\lambda_n$. We call this assertion the Extended Courant Property. In this paper, we propose new, simple and explicit examples for which the extended Courant property is false: convex domains in $\mathbb{R}^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$. We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.
Classification : 35P99, 35Q99, 58J50
Keywords: eigenfunction, nodal domain, Courant nodal domain theorem 
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     author = {B\'erard, Pierre H. and Helffer, Bernard},
     title = {On {Courant's} {Nodal} {Domain} {Property} for {Linear} {Combinations} of {Eigenfunctions.} {I}},
     journal = {Documenta mathematica},
     pages = {1561--1585},
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     volume = {23},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a20/}
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Bérard, Pierre H.; Helffer, Bernard. On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I. Documenta mathematica, Tome 23 (2018), pp. 1561-1585. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a20/