A negative result for hearing the shape of a triangle: a computer-assisted proof
Gerard Orriols Giménez · Dialnet (Universidad de la Rioja) · 2020
catalaEn aquest article demostrem que existeixen dos triangles diferents pels quals el primer, segon i quart valor propi del Laplacia amb condicions de Dirichlet coincideixen. Aixo resol una conjectura proposada per Antunes i Freitas i suggerida per la seva evidencia numerica. La prova es assistida per ordinador i utilitza una nova tecnica per tractar l’espectre d’un l’operador, que consisteix a combinar un Metode d’Elements Finits per localitzar aproximadament els primers valors propis i controlar la seva posicio a l’espectre, juntament amb el Metode de Solucions Particulars per confinar aquests valors propis a un interval molt mes precis. EnglishWe 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.