A negative result for hearing the shape of a triangle: a computer-assisted proof

Authors

  • Gerard Orriols Giménez ETH Zürich

Keywords:

computer-assisted proof, Laplace eigenvalues, spectral geometry, Finite Element Method, Method of Particular Solutions.

Abstract

We prove that there exist two distinct triangles for which the rst, second and fourth eigenvalues of the Laplace operator with zero Dirichlet boundary conditions coincide. This solves a conjecture raised by Antunes and Freitas and suggested by their numerical evidence. We use a novel technique for a computer-assisted proof about the spectrum of an operator, which combines a Finite Element Method, to locate roughly the rst eigenvalues keeping track of their position in the spectrum, and the Method of Particular Solutions, to get a much more precise bound of these eigenvalues.

Keywords: computer-assisted proof, Laplace eigenvalues, spectral geometry, Finite Element Method, Method of Particular Solutions.

Downloads

Download data is not yet available.

Downloads

How to Cite

Orriols Giménez, G. (2020). A negative result for hearing the shape of a triangle: a computer-assisted proof. Reports@SCM, 5(1), 33–44. Retrieved from https://revistes.iec.cat/index.php/reports/article/view/148601

Issue

Vol. 5 No. 1 (2020)

Section

Articles

License

The intellectual property of articles belongs to the respective authors.
On submitting articles for publication to the journal Reports@SCM, authors accept the following terms:
  1. Authors assign to Societat Catalana de Matemàtiques (a subsidiary of Institut d’Estudis Catalans) the rights of reproduction, communication to the public and distribution of the articles submitted for publication to Reports@SCM.
  2. Authors answer to Societat Catalana de Matemàtiques for the authorship and originality of submitted articles.
  3. Authors are responsible for obtaining permission for the reproduction of all graphic material included in articles.
  4. Societat Catalana de Matemàtiques declines all liability for the possible infringement of intellectual property rights by authors.
  5. The contents published in the journal, unless otherwise stated in the text or in the graphic material, are subject to a Creative Commons Attribution-NonCommercial-NoDerivs (by-nc-nd) 3.0 Spain licence, the complete text of which may be found at https://creativecommons.org/licenses/by-nc-nd/3.0/es/deed.en. Consequently, the general public is authorised to reproduce, distribute and communicate the work, provided that its authorship and the body publishing it are acknowledged, and that no commercial use and no derivative works are made of it.
  6. The journal Reports@SCM is not responsible for the ideas and opinions expressed by the authors of the published articles.