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

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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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Acknowledgements

The research of NN was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are grateful to Professor Arieh Iserles for helpful comments on this paper, which is dedicated to the occasion of his 70th birthday. His many mathematical contributions and his willingness to approach new problems with nonstandard techniques have deeply influenced this work.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Nilima Nigam

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Bartłomiej Siudeja & Benjamin Young

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  1. Nilima Nigam
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  2. Bartłomiej Siudeja
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  3. Benjamin Young
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Correspondence to Nilima Nigam.

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Communicated by Albert Cohen.

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Nigam, N., Siudeja, B. & Young, B. A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon. Found Comput Math 20, 1475–1504 (2020). https://doi.org/10.1007/s10208-020-09447-y

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