Stability and instability issues of the Weinstock inequality

Abstract

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