We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if such perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any generic assumption on the topology of the nodal sets or the simplicity of the Laplace spectrum. As an indirect application, we are able to show that a certain “Payne property” concerning the second nodal set remains stable under controlled perturbations of the domain.

中文翻译：

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小扰动下的拉普拉斯特征函数的节点集

我们研究了在特定小扰动下的紧凑流形上的拉普拉斯特征函数的节点集的稳定性。我们证明，如果这样的扰动相对较小，更正式地在亚波长范围内得到支持，则节点集将相当稳定。我们不需要关于节点集的拓扑或拉普拉斯光谱的简单性的任何一般性假设。作为间接的应用，我们能够证明与第二个节点集有关的某个“佩恩性质”在该域的受控扰动下保持稳定。