Title :

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

Author(s) :

Jollivet, Alexandre [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Sharafutdinov, Vladimir [Auteur]
Sobolev Institute of Mathematics

Journal title :

Journal of Geometric Analysis

Publication date :

2022

HAL domain(s) :

Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Théorie spectrale [math.SP]

English abstract : [en]

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 ...
Show more >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].Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL