Abstract
We address the question of determining the eigenvalues \({\lambda_{n}}\) (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with \({n}\) nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aronovitch A., Band R., Fajman D., Gnutzmann S.: Nodal domains of a non-separable domain—the right-angled isosceles triangle. J. Phys. A 45, 085209 (2012)
Band, R., Bersudsky, M., Fajman, D.: A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle. arXiv:1507.03410
Bérard P.: Spectres et groupes cristallographiques I: Domaines euclidiens. Invent. Math. 58, 179–199 (1980)
Bérard P., Besson G.: Spectres et groupes cristallographiques II: domaines sphériques. Ann. Inst. Fourier (Grenoble). 3, 237–248 (1980)
Bérard, P., Helffer, B.: Remarks on the boundary set of spectral equipartitions. Philos. Trans. A 372, 20120492 (2013). doi:10.1098/rsta.2012.0492
Bérard, P., Helffer, B.: Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses. In: Baklouti A., El Kacimi A., Kallel S., Mir, N., (eds.). Analysis and Geometry. MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi. Springer Proceedings in Mathematics and Statistics 127, pp. 69–114. Springer International Publishing (2015). arXiv:1402.6054
Bonnaillie-Noël, V., Helffer, B.: Nodal and spectral minimal partitions, The state of the art in 2015. To appear in the book “Shape optimization and spectral theory”, Henrot A., (ed.). arXiv:1506.07249. (in preparation)
Bourbaki N.: Groupes et algèbres de Lie, Chapitres 4, 5 et 6. Hermann, Paris (1968)
Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachr. Ges. Göttingen 81–84 (1923)
Rayleigh–Faber–Krahn inequality. Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php?title=Rayleigh-Faber-Krahn_inequality
Helffer, B., Persson-Sundqvist, M.: Nodal domains in the square–the Neumann case. To appear in Mosc. Math. J. arXiv:1410.6702
Hoffmann-Ostenhof T., Michor P.W., Nadirashvili N.: Bounds on the multiplicity of eigenvalues for fixed membranes. Geom. Funct. Anal. 9, 1169–1188 (1999)
Howards H., Hutchings M., Morgan F.: The isoperimetric problem on surfaces. Am. Math. Monthly 106, 430–439 (1999)
Keller J.B., Rubinow S.I.: Asymptotic solutions of eigenvalue problems. Ann. Phys. 9, 24–75 (1960)
Lamé G.: Mémoire sur la propagation de la chaleur dans les polyèdres. J. Éc. polytec. Math. 22, 194–251 (1833)
Léna, C.: Courant-sharp eigenvalues of a two-dimensional torus. arXiv:1501.02558
McCartin, B.J.: Laplacian eigenstructure of the equilateral triangle. Hikari Ltd (2011). http://www.m-hikari.com/mccartin-3
Pinsky M.A.: The eigenvalues of an equilateral triangle. SIAM J. Math. Anal. 11, 819–827 (1980)
Pinsky M.A.: Completeness of the eigenfunctions of the equilateral triangle. SIAM J. Math. Anal. 16, 848–851 (1985)
Pleijel A.: Remarks on Courant’s nodal theorem. Commun. Pure. Appl. Math. 9, 543–550 (1956)
Stern, A.: Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Inaugural-Dissertation zur Erlangung der Doktorwürde der Hohen Mathematisch-Naturwissenschaftlichen Fakultät der Georg August-Universität zu Göttingen (30 Juli 1924). Druck der Dieterichschen Univertisäts-Buchdruckerei (W. Fr. Kaestner). Göttingen (1925)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Louis Boutet de Monvel
Rights and permissions
About this article
Cite this article
Bérard, P., Helffer, B. Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle. Lett Math Phys 106, 1729–1789 (2016). https://doi.org/10.1007/s11005-016-0819-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-016-0819-9