Weinstock不等式的稳定性和不稳定性问题
Stability and instability issues of the Weinstock inequality
摘要：
给定两个平面共形光滑开集$Ω$和$ω$，证明了存在一个光滑集序列$Ω_n$，它几何收敛于$Ω$，并且使得$Ω_n$的（周长归一化）Steklov特征值收敛于$ω$的特征值.因此，我们回答了Girouard和Polterovich提出的一个关于Weinstock不等式稳定性的问题，并证明了这个不等式是真正不稳定的。然而，根据一些先验知识的几何形状有关的振荡的边界，稳定性可能会发生。
英文摘要：
Given two planar, conformal, smooth open sets $Ω$ and $ω$, we prove the existence of a sequence of smooth sets $Ω_n$ which geometrically converges to $Ω$ and such that the (perimeter normalized) Steklov eigenvalues of $Ω_n$ converge to the ones of $ω$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.
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