Monotone properties of the second even Neumann eigenfunction in symmetric domains

This paper investigates symmetric Neumann eigenfunctions on a class of planar domains that possess a single line of symmetry. We focus on the Neumann eigenfunctions corresponding to the second eigenvalue among functions that are symmetric with respect to the axis of symmetry of the domain. By employing the method of continuity, we establish the monotonicity of these eigenfunctions in the direction parallel to the axis of symmetry. Furthermore, we show that the corresponding eigenfunction is a Morse function on the boundary with exactly two critical points, and that the associated eigenvalue is simple within the space of symmetric functions.