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

Autors/ores

  • Gerard Orriols Giménez ETH Zürich

Paraules clau:

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

Resum

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.

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Com citar

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

Número

Vol. 5 Núm. 1 (2020)

Secció

Articles

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