Steklov rigidity of Euclidean balls

Abstract

Let $Ω\subset \mathbb{R}^n$, $n\geq 3$, be a bounded domain with smooth boundary. We show that if the Steklov spectrum of $Ω$ tends to that of a ball at a sufficiently fast rate, then $Ω$ must itself be a ball. In particular, in any dimension and among all bounded domains with smooth and possibly disconnected boundary, Euclidean balls are uniquely determined by their Steklov spectrum.