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Article Dans Une Revue Documenta Mathematica Année : 2018

On Courant's nodal domain property for linear combinations of eigenfunctions, Part I

Résumé

According to Courant's theorem, an eigenfunction as\-sociated 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 \emph{Extended Courant Property}.\smallskip In this paper, we propose simple and explicit examples for which the extended Courant property is false: convex domains in $\R^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$.
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Dates et versions

hal-01519629 , version 1 (08-05-2017)
hal-01519629 , version 2 (27-08-2017)
hal-01519629 , version 3 (19-10-2017)
hal-01519629 , version 4 (18-05-2018)
hal-01519629 , version 5 (19-10-2018)

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Pierre Bérard, Bernard Helffer. On Courant's nodal domain property for linear combinations of eigenfunctions, Part I. Documenta Mathematica, 2018, 23, pp.1561--1585. ⟨hal-01519629v5⟩
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