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

According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $λ\_n$ 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 $λ\_n$. We call this assertion the \emph{Extended Courant Property}.\smallskipIn 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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