Courant-sharp eigenvalues of the three-dimensional square torus

Léna, Corentin

doi:10.1090/proc/13148

Courant-sharp eigenvalues of the three-dimensional square torus
HTML articles powered by AMS MathViewer

by Corentin Léna

Proc. Amer. Math. Soc. 144 (2016), 3949-3958

DOI: https://doi.org/10.1090/proc/13148

Published electronically: April 27, 2016

PDF | Request permission

Abstract:

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb {R}/\mathbb {Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber–Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Pérez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.

References

R. Band, M. Bersudsky, and D. Fajman, A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle, ArXiv e-prints (2015).
P. Bérard, Inégalités isopérimétriques et applications. Domaines nodaux des fonctions propres, Goulaouic-Meyer-Schwartz Seminar, 1981/1982, École Polytech., Palaiseau, 1982, pp. Exp. No. XI, 10 (French). MR 671608
P. Bérard and B. Helffer, Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses, Analysis and Geometry (A. Baklouti, A. El Kacimi, S. Kallel, and N. Mir, eds.), Springer Proceedings in Mathematics & Statistics, vol. 127, Springer International Publishing, 2015, pp. 69–114 (English).
—, Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle, Lett. Math. Phys. (2016), 1–61.
Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513–541 (French). MR 690651, DOI 10.24033/asens.1435
Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
R. Courant, Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke., Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1923 (1923), 81–84 (German).
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
Laurent Hauswirth, Joaquín Pérez, Pascal Romon, and Antonio Ros, The periodic isoperimetric problem, Trans. Amer. Math. Soc. 356 (2004), no. 5, 2025–2047. MR 2031051, DOI 10.1090/S0002-9947-03-03362-2
B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 101–138. MR 2483815, DOI 10.1016/j.anihpc.2007.07.004
Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini, On spectral minimal partitions: the case of the sphere, Around the research of Vladimir Maz’ya. III, Int. Math. Ser. (N. Y.), vol. 13, Springer, New York, 2010, pp. 153–178. MR 2664708, DOI 10.1007/978-1-4419-1345-6_{6}
B. Helffer and R. Kiwan, Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality, ArXiv e-prints (2015).
B. Helffer and M. Persson Sundqvist, Nodal domains in the square—the Neumann case, Mosc. Math. J. 15 (2015), no. 3, 455–495.
—, On nodal domains in Euclidean balls, ArXiv e-prints (2015).
Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), no. 5, 430–439. MR 1699261, DOI 10.2307/2589147
Corentin Léna, Courant-sharp eigenvalues of a two-dimensional torus, C. R. Math. Acad. Sci. Paris 353 (2015), no. 6, 535–539 (English, with English and French summaries). MR 3348988, DOI 10.1016/j.crma.2015.03.014
Josef Leydold, On the number of nodal domains of spherical harmonics, Topology 35 (1996), no. 2, 301–321. MR 1380499, DOI 10.1016/0040-9383(95)00028-3
Åke Pleijel, Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math. 9 (1956), 543–550. MR 80861, DOI 10.1002/cpa.3160090324
Iosif Polterovich, Pleijel’s nodal domain theorem for free membranes, Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021–1024. MR 2457442, DOI 10.1090/S0002-9939-08-09596-8
Antonio Ros, The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260
John A. Toth and Steve Zelditch, Counting nodal lines which touch the boundary of an analytic domain, J. Differential Geom. 81 (2009), no. 3, 649–686. MR 2487604