Stability and instability issues of the Weinstock inequality

Bucur, Dorin; Nahon, Mickaël

doi:10.1090/tran/8302

Stability and instability issues of the Weinstock inequality
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by Dorin Bucur and Mickaël Nahon PDF

Trans. Amer. Math. Soc. 374 (2021), 2201-2223 Request permission

Abstract:

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

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