Stability and instability issues of the Weinstock inequality

Dorin Bucur; Mickaël Nahon

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arXiv · 2004.07784

Stability and instability issues of the Weinstock inequality

Dorin Bucur·Mickaël Nahon

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Abstract

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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Dorin Bucur, Mickaël Nahon. 2020-06-03. Stability and instability issues of the Weinstock inequality. https://arxiv.org/abs/2004.07784

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