Timeline for What's wrong with the Courant nodal domain theorem?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2023 at 13:10 | answer | added | Ruifeng Chen | timeline score: 1 | |
| Mar 9, 2023 at 14:14 | comment | added | stewori | @GrahamCox Chavel writes "C. deVerdiere has noted that the regularity argument of Cheng has a gap when n = dim M > 2 and that one can even give a counterexample to the argument (though not, necessarily, the result) as it stands (see P. Bérard, D. Meyer - Inégalités isopérimétriques et applications)." The cited work is available on open access at numdam.org/article/ASENS_1982_4_15_3_513_0.pdf (unfortunately I cannot read French) | |
| Mar 9, 2023 at 13:58 | comment | added | stewori | @Kelei Wang As I understand the construction, the constructed function is not necessarily differentiable on the nodal set (at the border between nodal domains). How can it then obey the unique continuation principle? (In Courant's proof that step is more complicated. Can it be as simple as you say?) | |
| May 27, 2019 at 14:47 | history | edited | user64494 | CC BY-SA 4.0 |
A typo in the title is corrected.
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| Feb 8, 2017 at 22:11 | comment | added | Graham Cox | Could you elaborate on your comment that Cheng's proof is incomplete (or provide a reference)? | |
| Mar 14, 2015 at 1:07 | vote | accept | Fan Zheng | ||
| Jan 7, 2014 at 4:26 | comment | added | Kelei Wang | But the argument proceeds as follows. Assume the number of nodal domains of the $k$th eigenfunction $u_k$ is larger than $k$, then you can find a function vanishing on a nodal domain, orthogonal to the first $k-1$ eigenfuncions and attaining the $k$th eigenvalue. Thus by the min-max principle it's an eigenfunction. But it vanishes on an open set, thus must be $0$ by the unique continuation principle. This argument does not involve any regularity of the nodal sets. | |
| Jan 5, 2014 at 18:18 | comment | added | guest | @kelei no you are probably thinking of the courant-fischer-weyl min-max principle en.wikipedia.org/wiki/Min-max_theorem. The nodal domain theorem is about the characterization of zeros of eigenfunctions, not so much about the characterization of eigenvalues. | |
| Dec 6, 2013 at 17:34 | answer | added | user42070 | timeline score: 10 | |
| Nov 29, 2013 at 1:58 | comment | added | Kelei Wang | Courant nodal domain theorem only involves the max-min characterization of eigenvalues. | |
| Nov 28, 2013 at 16:14 | history | asked | Fan Zheng | CC BY-SA 3.0 |