Nearly radial Neumann eigenfunctions on symmetric domains

TL;DR

This paper investigates the existence of Neumann eigenfunctions that are positive on the boundary of symmetric domains, demonstrating their existence on polygons with at least 5 sides and developing numerical methods for complex cases.

Contribution

It introduces a new combination of analytic and numerical methods to establish the existence of positive boundary eigenfunctions on specific symmetric domains, including regular polygons and triangles.

Findings

Eigenfunctions positive on boundary exist for polygons with ≥5 sides.

No such eigenfunctions exist for equilateral triangles and cubes.

Validated numerical methods confirm results for complex cases.

Abstract

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.