Geodesic
On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I
Documenta mathematica, Tome 23 (2018), pp. 1561-1585
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According to Courant's theorem, an eigenfunction associated with the n-th eigenvalue λn​ 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 λn​. 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 Rn (hypercube and equilateral triangle), domains with cracks in R2, on the round sphere S2, and on a flat torus T2. We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.
DOI : 10.4171/dm/652
Classification : 35P99, 35Q99, 58J50
Mots-clés : nodal domain, eigenfunction, Courant nodal domain theorem 
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     author = {Bernard Helffer and Pierre H. B\'erard},
     title = {On {Courant's} {Nodal} {Domain} {Property} for {Linear} {Combinations} of {Eigenfunctions.} {I}},
     journal = {Documenta mathematica},
     pages = {1561--1585},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/652},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/652/}
}
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Bernard Helffer; Pierre H. Bérard. On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I. Documenta mathematica, Tome 23 (2018), pp. 1561-1585. doi: 10.4171/dm/652

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