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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
Article

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
$$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$
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.

Keywords

Nodal Domain Hexagonal Packing Disk Packing Minimal Partition Hexagonal Tiling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutBonnGermany

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