A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon

Abstract

A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartlomiej Siudeja it was shown 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. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present 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. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.