Consider the first eigenfunction of the Dirichlet Laplacian in a domain which looks like a cross pommy, see the figure below. Since the domain is symmetric, so does the first eigenfunction. In particular, it should have a critical point at the center of the domain. Which type of critical point is it?
At first glance, one might think that it is a saddle point, especially if the channels are very thin. However, it must be a point of local maximum. Indeed, if we suppose that it is a saddle point, then it must be degenerate due the symmetry of the eigenfunction. Thus, the Laplacian at this point has to be zero, which contradicts the positivity of the first eigenfunction.
This fact also implies that there has to be at least 4 saddle points nearby the center. This observation is correctly reflected in numerical calculations, see the figure below.
Last modified: 18-Sep-25