A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate
A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate
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几何不确定性原理及其在 Pleijel 估计中的应用
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发表时间：
2013
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通讯作者：
S. Steinerberger
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作者：
S. Steinerberger
摘要设 $${\Omega \subset \mathbb{R}^2}$$Ω⊂R2 为开有界域，$${\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}$$Ω=⋃i=1NΩi 为分区。将弗拉恩克尔不对称性表示为 $${0 \leq \mathcal{A}(\Omega_i) \leq 2}$$0≤A(Ωi)≤2 并写成
$$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$D(Ωi):=|Ωi|-min1≤j≤N|Ωj||Ωi|，其中 $${0 \leq D(\Omega_{i}) \leq 1}$$0≤D(Ωi)≤1。对于仅取决于 $${\Omega}$$Ω 足够大的 N，存在不确定性原理
$$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$Σi=1N|Ωi||Ω|A(Ωi)+Σi=1N|Ωi||Ω|D(Ωi)≥160000。对于某个常数 $${c_{n} > 0}$$cn>0，该语句在维度 $${n \geq 3}$$n≥3 中仍然成立。作为一个应用，我们对 Pleijel 对拉普拉斯本征函数节点域数量的估计进行了（未指定的）改进，并改进了谱划分问题的不等式。
AbstractLet $${\Omega \subset \mathbb{R}^2}$$Ω⊂R2 be an open, bounded domain and $${\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}$$Ω=⋃i=1NΩi be a partition. Denote the Fraenkel asymmetry by $${0 \leq \mathcal{A}(\Omega_i) \leq 2}$$0≤A(Ωi)≤2 and write
$$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$D(Ωi):=|Ωi|-min1≤j≤N|Ωj||Ωi|with $${0 \leq D(\Omega_{i}) \leq 1}$$0≤D(Ωi)≤1. For N sufficiently large depending only on $${\Omega}$$Ω, there is an uncertainty principle
$$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$∑i=1N|Ωi||Ω|A(Ωi)+∑i=1N|Ωi||Ω|D(Ωi)≥160000.The statement remains true in dimensions $${n \geq 3}$$n≥3 for some constant $${c_{n} > 0}$$cn>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.