论文标题
温斯托克不平等的稳定性和不稳定性问题
Stability and instability issues of the Weinstock inequality
论文作者
论文摘要
给定两个平面，共形，平滑的打开集$ω$和$ω$，我们证明存在一系列平滑序列$ω_n$，几何粒子收敛到$ω$，并且（周范围的）steklov eigenvalues $ω_n$ to $ω_n$会收敛到$ω$的$ω$。结果，我们回答了Girouard和Polterovich提出的关于Weinstock不平等的稳定性提出的问题，并证明不平等是真正不稳定的。但是，根据一些与边界振荡有关的几何形状的先验知识，可能会发生稳定性。
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.
