On Courant’s nodal domain property for linear combinations of eigenfunctions, Part I Article - Octobre 2018

Pierre Bérard, Bernard Helffer

Pierre Bérard, Bernard Helffer, « On Courant’s nodal domain property for linear combinations of eigenfunctions, Part I  », Documenta Mathematica, octobre 2018, pp. 1561—1585. ISSN 1431-0643. 〈https://www.elibm.org/article/10011882〉

Abstract

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 \emphExtended 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 $\mathbbR^2$, on the round sphere $\mathbbS^2$, and on a flat torus $\mathbbT^2$.

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