Nearly radial Neumann eigenfunctions on symmetric domains

Abstract

We study the existence of Neumann eigenfunctions of the Laplacian 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 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. This work builds on recent results of Hoffmann–Ostenhof about rectangles.

We use analytic methods combined with symmetry arguments to prove the result for polygons with six or more sides, and combinatorics for equilateral triangles and cubes. The case of the regular pentagon is much harder. Its proof requires deep computational and numerical results which are beyond the scope of the present paper. The pentagonal case, codeveloped with Nilima Nigam and Benjamin Young, will appear in a companion paper.