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.



References for Neumann eigenfunctions
From this recent [paper](https://arxiv.org/abs/1802.01800) 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.](http://michaelnielsen.org/polymath1/index.php?title=The_hot_spots_conjecture)


