Abstract
In this paper, we revisit Courant’s nodal domain theorem for the Dirichlet eigenfunctions of a square membrane, and the analyses of A. Stern and Å. Pleijel.
In memory of M. Salah Baouendi
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Notes
- 1.
English translation from German citation:
[E1] ... In dimension one, according to Sturm’s theorem (see footnote 2), the interval is divided into n subsets by the nodes of the nth eigenfunction. This rule no longer holds for multidimensional problems, ... It can be easily shown that on the sphere, for each eigenvalue, two or three appear as numbers of nodal domains, and that when ordering the eigenvalues in nondecreasing order the number of nodal domains 2 always reappears.
[Q1] ... We now want to show that for the square the number of nodal domains two always reappears.
[K1] ... We next want to show that for each eigenvalue there exists an eigenfunction whose nodal lines divide the sphere into two or three domains.
[K2] ... the number of nodal domains two appears for all eigenvalues \(\lambda _n = (2r+1)(2r+2)\qquad r=,1,2,\ldots \) and we now want to show that the number of nodal domains three always reappears for all eigenvalues
$$ \lambda _n =2r (2r+1)\qquad r=1,2,\ldots $$ - 2.
Journal de Mathématiques, T.1, 1836, pp. 106–186, 269–277, 375–444.
- 3.
English translation from German citation:
[Q2] ... We consider the eigenvalues
$$ \lambda _n= \lambda _{2r,1} = 4r^2 +1\,,\, r=1,2,\dots $$and the nodal lines of the associated eigenfunction
$$ u_{2r,1} + u_{1,2r} =0\,, $$which, as can be easily proved by using graphic images, gives Fig. 7.
[Q3] ... If starting from \(\mu =1\) we decrease \(\mu \), then the double points of the nodal lines disappear simultaneously and in the same way as shown in Fig. 8. As the nodal set consists of one line without double point, the square becomes divided into two domains and this occurs for all eigenvalues \(\lambda _{n} = \lambda _{2r,1}=4r^2+1\,\) (\(r=1,2,\dots )\).
- 4.
English translation from German citation:
[I1 ... In order to determine the typical course of the nodal lines, we have similar key points as for the sphere. If we superimpose the systems of nodal lines of \(u_{\ell ,m}\) and \(u_{m,\ell }\), then for \(\mu >0\) (\(<0\)) the nodal lines can only visit the domains for which the two functions have opposite (same) signs.
[I2] ... Moreover all the nodal lines associated with the eigenvalue \(\lambda _{\ell ,m}\) should meet all the crossing points between the nodal sets \(u_{\ell ,m}=0\) and \(u_{m,\ell }=0\), hence going through \((\ell -1)^2 + (m-1)^2\) fixed points ...
- 5.
‘Schraffieren’, see [20, tag I1], in the spherical case.
- 6.
A. Stern and H. Lewy were both students of R. Courant at about the same time, 1925. H. Lewy does however not refer to A. Stern’s Thesis in his paper. We refer to [3] for a further discussion.
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Bérard, P., Helffer, B. (2015). Dirichlet Eigenfunctions of the Square Membrane: Courant’s Property, and A. Stern’s and Å. Pleijel’s Analyses. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir, N. (eds) Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-319-17443-3_6
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