Annales Henri Poincaré

, Volume 15, Issue 12, pp 2299–2319 | Cite as

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

  • Stefan SteinerbergerEmail author


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
$$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$
with \({0 \leq D(\Omega_{i}) \leq 1}\). For N sufficiently large depending only on \({\Omega}\), there is an uncertainty principle
$$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$
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.


Nodal Domain Hexagonal Packing Disk Packing Minimal Partition Hexagonal Tiling 
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© Springer Basel 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutBonnGermany

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