Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove the existence of a sequence of smooth sets $\Omega_n$ which geometrically converges to $\Omega$ and such that the (perimeter normalized) Steklov eigenvalues of $\Omega_n$ converge to the ones of $\omega$. 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.

中文翻译：

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温斯托克不等式的稳定性和不稳定性问题

给定两个平面、共形、光滑的开集 $\Omega$ 和 $\omega$，我们证明了一系列光滑集 $\Omega_n$ 的存在，它在几何上收敛到 $\Omega$ 并且使得（周长归一化）Steklov $\Omega_n$ 的特征值收敛到 $\omega$ 的特征值。因此，我们回答 Girouard 和 Polterovich 提出的关于 Weinstock 不等式稳定性的问题，并证明该不等式确实不稳定。然而，在与边界振荡相关的几何形状的一些先验知识下，可能会出现稳定性。