Abstract
Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write
with \({0 \leq D(\Omega_{i}) \leq 1}\). For N sufficiently large depending only on \({\Omega}\), there is an uncertainty principle
The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\). As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.
Article PDF
Similar content being viewed by others
References
Bérard, P.: Inégalités isopérimétriques et applications. Domaines nodaux des fonctions propres. Goulaouic-Meyer-Schwartz Seminar, 1981/1982, Exp. No. XI, 10 pp., École Polytech., Palaiseau (1982)
Bérard, P., Helffer, B.: Remarks on the Boundary Set of Spectral Equipartitions. arXiv:1203.3566
Blind G.: Über Unterdeckungen der Ebene durch Kreise. J. Reine Angew. Math. 236, 145–173 (1969)
Blum G., Gnutzmann S., Smilansky U.: Nodal domains statistics: a criterion for quantum chaos. Phys. Rev. Lett. 88, 114101 (2002)
Bourgain, J.: On Pleijel’s Nodal Domain Theorem. arxiv.org:1308.4422
Brasco, L., De Philippis, G., Velichkov, B.: Faber-Krahn inequalities in sharp quantitative form. arXiv:1306.0392
Cafferelli L., Lin F.H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1–2), 5–18 (2007)
Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachr. Ges. Göttingen, pp. 81–84 (1923)
Fusco N., Maggi F., Pratelli A.: Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(1), 51–71 (2009)
Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168, 941–980 (2008)
Helffer B.: On spectral minimal partitions: a survey. Milan J. Math. 78(2), 575–590 (2010)
Helffer B., Hoffmann-Ostenhof T., Terracini S.: Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 101–138 (2009)
Jia-Chang S.: On approximation of Laplacian eigenproblem over a regular hexagon with zero boundary conditions. Special issue dedicated to the 70th birthday of Professor Zhong-Ci Shi. J. Comput. Math. 22(2), 275–286 (2004)
Pleijel A.: Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math. 9, 543–550 (1956)
Polterovich I.: Pleijel’s nodal domain theorem for free membranes. Proc. Am. Math. Soc. 137(3), 1021–1024 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jens Marklof.
The author is grateful for various discussions about spectral partition problems with Bernhard Helffer.
Rights and permissions
About this article
Cite this article
Steinerberger, S. A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate. Ann. Henri Poincaré 15, 2299–2319 (2014). https://doi.org/10.1007/s00023-013-0310-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0310-4