Bobkov, Vladmir

On Exact Pleijel's Constant for Some Domains

Doc. Math. 23, 799-813 (2018)
DOI: 10.25537/dm.2018v23.799-813
Communicated by Heinz Siedentop


We provide an explicit expression for the Pleijel constant for the planar disk and some of its sectors, as well as for $N$-dimensional rectangles. In particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we characterize the Pleijel constant for some rings and annular sectors in terms of asymptotic behavior of zeros of certain cross-products of Bessel functions.

Mathematics Subject Classification

35P05, 35P20, 35B05, 35J05


Pleijel theorem, eigenvalues, nodal domain, Courant's nodal domain theorem, Bessel functions


  • 1. Aronovitch, A., Band, R., Fajman, D. & Gnutzmann, S. (2012). Nodal domains of a non-separable problem - the right-angled isosceles triangle. Journal of Physics A: Mathematical and Theoretical, 45(8), 085209. href; zbl 1248.35131; MR2897017; arXiv:href;; DOI 10.1088/1751-8113/45/8/085209nolinkurlDOI:10.1088/1751-8113/45/8/085209; arxiv org/abs/1110.15211110.1521.
  • 2. Ashu, A. M. (2013). Some properties of Bessel functions with applications to Neumann eigenvalues in the unit disc. Bachelor's thesis (E. Wahlén advisor), Lund University.
  • 3. Bérard, P. & Meyer, D. (1982). Inégalités isopérimétriques et applications. Annales scientifiques de l'École Normale Supérieure, 15(3), 513-541.; zbl 0527.35020; MR0690651.
  • 4. Blum, G., Gnutzmann, S. & Smilansky, U. (2002). Nodal domains statistics: A criterion for quantum chaos. Physical Review Letters, 88(11), 114101. href arXiv:href; DOI 10.1103/PhysRevLett.88.114101nolinkurlDOI:10.1103/PhysRevLett.88.114101; arxiv org/abs/nlin/01090290109029.
  • 5. Bonnaillie-Noël, V., Helffer, B. & Hoffmann-Ostenhof, T. (2017). Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators. IAMP News Bulletin, October 2017, 3-28. arXiv:href; arxiv org/abs/1711.011741711.01174.
  • 6. Bonnaillie-Noël, V. & Léna, C. (2014). Spectral minimal partitions of a sector. Discrete and Continuous Dynamical Systems-Series B, 19(1), 27-53. href; zbl 1286.35175; MR3245081; DOI 10.3934/dcdsb.2014.19.27nolinkurlDOI:10.3934/dcdsb.2014.19.27.
  • 7. Bourgain, J. (2015). On Pleijel's nodal domain theorem. International Mathematics Research Notices, 2015(6), 1601-1612. href; zbl 1317.35145; MR3340367; arXiv:href; MRn/rnt241nolinkurl; arxiv org/abs/1308.44221308.4422.
  • 8. Charron, P. (2018). A Pleijel-type theorem for the quantum harmonic oscillator. Journal of Spectral Theory, 8(2), 715-732. href; zbl 06898063; MR3812813; arXiv:href; DOI 10.4171/JST/211nolinkurlDOI:10.4171/JST/211; arxiv org/abs/1512.078801512.07880.
  • 9. Cochran, J. A. (1964). Remarks on the zeros of cross-product Bessel functions. Journal of the Society for Industrial and Applied Mathematics, 12(3), 580-587. href; zbl 0132.05601; MR0178169; DOI 10.1137/0112049nolinkurlDOI:10.1137/0112049.
  • 10. Cochran, J. A. (1966). The analyticity of cross-product Bessel function zeros. Proceedings of the Cambridge Philosophical Society, 62(2), 215-226. href; zbl 0135.28003; MR0197794; DOI 10.1017/S0305004100039785nolinkurlDOI:10.1017/S0305004100039785.
  • 11. Courant, R. (1923). Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 81-84.; jfm 49.0342.01.
  • 12. Elbert, Á. (2001). Some recent results on the zeros of Bessel functions and orthogonal polynomials. Journal of computational and applied mathematics, 133(1-2), 65-83. href; zbl 0989.33004; MR1858270; DOI 10.1016/S0377-0427(00)00635-XnolinkurlDOI:10.1016/S0377-0427(00)00635-X.
  • 13. Elbert, Á. & Laforgia, A. (1994). A lower bound for the zeros of the Bessel functions. In Inequalities And Applications (pp. 179-185). href; zbl 0900.33008; MR1299553; DOI 10.1142/9789812798879_0015nolinkurlDOI:10.1142/9789812798879_0015.
  • 14. Gnutzmann, S. & Lois, S. (2013). On the nodal count statistics for separable systems in any dimension. Journal of Physics A: Mathematical and Theoretical, 46(4), 045201. href; zbl 1290.35165; MR3008781; arXiv:href; DOI 10.1088/1751-8113/46/4/045201nolinkurlDOI:10.1088/1751-8113/46/4/045201; arxiv org/abs/1208.21201208.2120.
  • 15. Han, X., Murray, M. & Tran, C. (2017). Nodal lengths of eigenfunctions in the disc. arXiv:href; arxiv org/abs/1708.081121708.08112.
  • 16. Helffer, B. & Hoffmann-Ostenhof, T. (2015). A review on large $k$ minimal spectral $k$-partitions and Pleijel's Theorem. Spectral theory and partial differential equations, 39-57, Contemp. Math., 640, Amer. Math. Soc., Providence, RI, 2015. href; zbl 1346.35132; MR3381015; arxiv 1509.04501; DOI 10.1090/conm/640/12841nolinkurlDOI:10.1090/conm/640/12841.
  • 17. Helffer, B. & Sundqvist, M. (2016). On nodal domains in Euclidean balls. Proceedings of the American Mathematical Society, 144(11), 4777-4791. href arXiv:href; DOI 10.1090/proc/13098nolinkurlDOI:10.1090/proc/13098; arxiv org/abs/1506.040331506.04033.
  • 18. Ivrii, V. Y. (1980). Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary. Functional Analysis and Its Applications, 14(2), 98-106. href; zbl 0453.35068; MR0575202; DOI 10.1007/BF01086550nolinkurlDOI:10.1007/BF01086550.
  • 19. Kershaw, D. (1983). Some extensions of W. Gautschi's inequalities for the gamma function. Mathematics of Computation, 607-611. href; zbl 0536.33002; MR0717706; DOI 10.2307/2007697nolinkurlDOI:10.2307/2007697.
  • 20. Kline, M. (1948). Some Bessel equations and their applications to guide and cavity theory. Studies in Applied Mathematics, 27(1-4), 37-48. href; MR0023961; DOI 10.1002/sapm194827137nolinkurlDOI:10.1002/sapm194827137.
  • 21. Léna, C. (2016). Pleijel's nodal domain theorem for Neumann eigenfunctions. arXiv:href; arxiv org/abs/1609.023311609.02331.
  • 22. McCann, R. C. (1977). Lower bounds for the zeros of Bessel functions. Proceedings of the American Mathematical Society, 64(1), 101-103. href; zbl 0364.33009; MR0442316; DOI 10.1090/S0002-9939-1977-0442316-6nolinkurlDOI:10.1090/S0002-9939-1977-0442316-6.
  • 23. McMahon, J. (1894). On the roots of the Bessel and certain related functions. The Annals of Mathematics, 9(1/6), 23-30. href; jfm 25.0842.02; MR1502177; DOI 10.2307/1967501nolinkurlDOI:10.2307/1967501.
  • 24. Pleijel, A. (1956). Remarks on Courant's nodal line theorem. Communications on pure and applied mathematics, 9(3), 543-550. href; zbl 0070.32604; MR0080861; DOI 10.1002/cpa.3160090324nolinkurlDOI:10.1002/cpa.3160090324.
  • 25. Polterovich, I. (2009). Pleijel's nodal domain theorem for free membranes. Proceedings of the American Mathematical Society, 137(3), 1021-1024. href; zbl 1162.35005; MR2457442; arXiv:href; DOI 10.1090/S0002-9939-08-09596-8nolinkurlDOI:10.1090/S0002-9939-08-09596-8; arxiv org/abs/0805.15530805.1553.
  • 26. Steinerberger, S. (2014). A Geometric Uncertainty Principle with an Application to Pleijel's Estimate. Annales Henri Poincaré, 15(12), 2299-2319. href; zbl 1319.35132; MR3272823; arXiv:href; DOI 10.1007/s00023-013-0310-4nolinkurlDOI:10.1007/s00023-013-0310-4; arxiv org/abs/1306.31031306.3103.
  • 27. Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge: The University Press. zbl 0063.08184; MR0010746.


Bobkov, Vladmir
Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic