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Published in: Foundations of Computational Mathematics 6/2020

19-02-2020

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

Authors: Nilima Nigam, Bartłomiej Siudeja, Benjamin Young

Published in: Foundations of Computational Mathematics | Issue 6/2020

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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 Bartłomiej 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.
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Metadata
Title
A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon
Authors
Nilima Nigam
Bartłomiej Siudeja
Benjamin Young
Publication date
19-02-2020
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 6/2020
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-020-09447-y

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