AN ESTIMATE FOR THE STEKLOV ZETA FUNCTION OF A PLANAR DOMAIN DERIVED FROM A FIRST VARIATION FORMULA

Alexandre Jollivet (1) , Vladimir Sharafutdinov (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Sobolev Institute of Mathematics

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].

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Alexandre Jollivet, Vladimir Sharafutdinov. AN ESTIMATE FOR THE STEKLOV ZETA FUNCTION OF A PLANAR DOMAIN DERIVED FROM A FIRST VARIATION FORMULA. Journal of Geometric Analysis, 2022. ⟨hal-02515278⟩