Nearly radial Neumann eigenfunctions on symmetric domains

We study the existence of Neumann eigenfunctions which do not change sign on the boundary of some special domains. We show that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 5 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. We use analytic methods combined with symmetry arguments to prove the result for polygons with six or more sides. The case for the regular pentagon is harder. We develop a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software.