Calculus of Variations and Geometric Measure Theory

D. Bucur - M. Nahon

Stability and instability issues of the Weinstock inequality

created by bucur on 16 Apr 2020


Submitted Paper

Inserted: 16 apr 2020
Last Updated: 16 apr 2020

Year: 2020


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.