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I am looking for references on eigenfunctions with Neumann boundary condition.

In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it have critical points at vertices. But, I could not find the source. I think this is probably a well-known fact. But I am very interested in the proof.

It may be a famous fact for experts, but I would appreciate it if you could tell me.

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    $\begingroup$ This is quite straightforward once we know that the gradient is well-defined and continuous. Indeed: by the Neumann boundary condition, the gradient of an eigenfunction at a boundary point is parallel to the boundary. Therefore, at a vertex, the gradient is parallel to two different directions, and hence it is necessarily zero. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Feb 15, 2020 at 21:33
  • $\begingroup$ @MateuszKwaśnicki Thank you very much for your comments. I am convinced. I didn't think of it until you said. The comments of Carlo Beenakker is in a different direction, right? $\endgroup$
    – sharpe
    Commented Feb 15, 2020 at 21:42
  • $\begingroup$ I think so, it deals with non-vertex critical points. Still, it is an excellent answer! $\endgroup$
    – Mateusz Kwaśnicki
    Commented Feb 15, 2020 at 21:44
  • $\begingroup$ @MateuszKwaśnicki I think so too! $\endgroup$
    – sharpe
    Commented Feb 15, 2020 at 21:46

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From this recent paper I would conclude the statement is false: the second Neumann eigenfunction of an acute triangle has one non-vertex critical point.

This was a Polymath problem.

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  • $\begingroup$ Thank you for your comment. Do you mean that the second Neumann eigenfunction on an acute triangle does not have critical points at vertices? $\endgroup$
    – sharpe
    Commented Feb 15, 2020 at 20:52
  • $\begingroup$ that is not how I understand the cited paper: the statement is that there is exactly one critical point that is not at a vertex, I don't think critical points at vertices are excluded. $\endgroup$
    – Carlo Beenakker
    Commented Feb 15, 2020 at 22:04

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