2020/03/24 by Jollivet, Alexandre, Sharafutdinov, Vladimir
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2004.01779
We consider the Steklov zeta function ζ Ω of a smooth bounded simply connected planar domain Ω ⊂ R 2 of perimeter 2π. We provide a first variation formula for ζ Ω under a smooth deformation of the domain. On the base of the formula, we prove that, for every s ∈ (--1, 0) ∪ (0, 1), the difference ζ Ω (s) -- 2ζ R (s) is non-negative and is equal to zero if and only if Ω is a round disk (ζ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality ζ Ω (s) -- 2ζ R (s) ≥ 0 for s ∈ (--∞, --1] ∪ (1, ∞); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality ζ Ω (0) = 2ζ R (0) obtained by Edward and Wu [1991].