This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem$$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$in two-dimensional domains \(\Omega \). In particular, this paper presents a dense family \(\mathcal {A}\) of simply-connected two-dimensional domains with analytic boundaries such that, for each \(\Omega \in \mathcal {A}\), the nodal set of the eigenfunction \(\phi _{\sigma _j}\) “is not dense at scale \(\sigma _j^{-1}\)”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains \(\Omega \in \mathcal {A}\), the nodal sets of the eigenfunctions \(\phi _{\sigma _j}\) associated with the eigenvalue \(\sigma _j\) have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each \(\Omega \in \mathcal {A}\) there is a value \(r_1>0\) such that for each j there is \(x_j\in \Omega \) such that \(\phi _{\sigma _j}\) does not vanish on the ball of radius \(r_1\) around \(x_j\).

中文翻译：

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没有密集Steklov节点集的域

本文涉及Steklov特征值问题本征函数的节点集的渐近几何特征$$ \ begin {aligned}-\ Delta \ phi _ {\ sigma _j} = 0，\ quad \ hbox {on} \，\ ，\ Omega，\ quad \ partial _ \ nu \ phi _ {\ sigma _j} = \ sigma _j \ phi _ {\ sigma _j} \ quad \ hbox {on} \，\，\ partial \ Omega \ end {aligned } $$位于二维域\（\ Omega \）中。特别是，本文提出了具有解析边界的简单连接二维域的密集族\（\ mathcal {A} \），使得对于每个\（\ omega \ in \ mathcal {A} \），节点集本征函数的\（\披_ {\西格玛_j} \） “是不致密的大规模\（\西格玛_j ^ { - 1} \）”。该结果解决了Girouard和Polterovich在“开放问题10”下提出的问题（J Spectr Theory 7（2）：321–359，2017）。实际上，本文的结果证明，对于域\（\ Omega \ in \ mathcal {A} \），特征函数\（\ phi _ {\ sigma _j} \）的节点集与特征值相关\（\ sigma _j \）具有与预期完全不同的特征：它们在任何缩小的尺度上都不致密。更确切地说，对于每个\（\ Omega \ in \ mathcal {A} \），都有一个值\（r_1> 0 \），使得对于每个j都具有\（x_j \ in \ Omega \），使得\（\ phi _ {\ sigma _j} \）在半径球上不消失\（r_1 \）在\（x_j \）周围。