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

• Gerard Orriols Giménez ^[1]
  1. [1] Swiss Federal Institute of Technology in Zurich

     Swiss Federal Institute of Technology in Zurich

     Zürich, Suiza

     
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 5, Nº. 1, 2020, págs. 33-44
• Idioma: inglés
• DOI: 10.2436/20.2002.02.21
• Enlaces
  • Texto completo
• Resumen
  • català

    En aquest article demostrem que existeixen dos triangles diferents pels quals el primer, segon i quart valor propi del Laplacià amb condicions de Dirichlet coincideixen. Això resol una conjectura proposada per Antunes i Freitas i suggerida per la seva evidència numèrica. La prova és assistida per ordinador i utilitza una nova tècnica per tractar l’espectre d’un l’operador, que consisteix a combinar un Mètode d’Elements Finits per localitzar aproximadament els primers valors propis i controlar la seva posició a l’espectre, juntament amb el Mètode de Solucions Particulars per confinar aquests valors propis a un interval molt més precís.

  • English

    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.

• Referencias bibliográficas
  • P.R.S. Antunes, P. Freitas. On the inverse spectral problem for Euclidean triangles, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130)...
  • T.P. Branson, P.B. Gilkey. The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15(2) (1990),...
  • A.H. Barnett, A. Hassell. Boundary quasiorthogonality and sharp inclusion bounds for large Dirichlet eigenvalues, SIAM J. Numer. Anal. 49(3)...
  • T. Betcke, L.N. Trefethen. Reviving the method of particular solutions, SIAM Rev. 47(3) (2005), 469–491.
  • P.-K. Chang, D. DeTurck. On hearing the shape of a triangle, Proc. Amer. Math. Soc. 105(4) (1989), 1033–1038.
  • C. Carstensen, J. Gedicke. Guaranteed lower bounds for eigenvalues, Math. Comp. 83(290) (2014), 2605–2629.
  • C. Durso. On the inverse spectral problem for polygonal domains, Ph.D. thesis, Massachusetts Institute of Technology, 1988.
  • L. Fox, P. Henrici, C. Moler. Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal. 4(1) (1967), 89–102.
  • D. Grieser, S. Maronna. Hearing the shape of a triangle, Notices Amer. Math. Soc. 60(11) (2013), 1440–1447.
  • J. G´omez-Serrano. Computer-assisted proofs in PDE: a survey, SeMA J. 76(3) (2019), 459–484.
  • J. G´omez-Serrano, G. Orriols. Any three eigenvalues do not determine a triangle, to appear in J. Differential Equations. Available at arXiv:1911.06758.
  • A. Gopal, L.N. Trefethen. Solving Laplace problems with corner singularities via rational functions, SIAM J. Numer. Anal. 57(5) (2019), 2074–2094.
  • C. Gordon, D.L. Webb, S. Wolpert. One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27(1) (1992), 134–138.
  • A. Henrot. Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkh¨auser Verlag, Basel, 2006.
  • F. Johansson. Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66(8) (2017), 1281–1292.
  • M. Kac. Can one hear the shape of a drum? Amer. Math. Monthly 73(4), part II (1966), 1–23.
  • X. Liu. A framework of verified eigenvalue bounds for self-adjoint differential operators, Appl. Math. Comput. 267 (2015), 341–355.
  • X. Liu, S. Oishi. Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal. 51(3) (2013),...
  • C. Miranda. Un’osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2) 3 (1940), 5–7.
  • H.P. McKean, Jr., I.M. Singer. Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1(1) (1967), 43–69.
  • B.N. Parlett. The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs,...
  • F. Rellich. Darstellung der Eigenwerte von ∆u+λu = 0 durch ein Randintegral, Math. Z. 46 (1940), 635–636.
  • W. Tucker. Validated numerics. A short introduction to rigorous computations, Princeton University Press, Princeton, NJ, 2011.
  • M. van den Berg, S. Srisatkunarajah. Heat equation for a region in R 2 with a polygonal boundary, J. London Math. Soc. (2) 37(1) (1988), 119–127.
  • S. Zelditch. Inverse spectral problem for analytic domains. II. Z2-symmetric domains, Ann. of Math. (2) 170(1) (2009), 205–269.