AN ESTIMATE FOR THE STEKLOV ZETA FUNCTION OF A PLANAR DOMAIN DERIVED FROM A FIRST VARIATION FORMULA - Archive ouverte HAL Access content directly
Journal Articles Journal of Geometric Analysis Year : 2022

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

Alexandre Jollivet
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Vladimir Sharafutdinov
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Abstract

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].
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Dates and versions

hal-02515278 , version 1 (23-03-2020)

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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⟩
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