Abstract: We consider the Steklov zeta function ζΩ of a smooth bounded simply connected planar domain Ω⊂ R2 of perimeter 2 π. We provide a first variation formula for ζΩ 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 ζΩ(s) - 2 ζR(s) is non-negative and is equal to zero if and only if Ω is a round disk (ζR is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality ζΩ(s) - 2 ζR(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 ζΩ′(0)=2ζR′(0) obtained by Edward and Wu (Determinant of the Neumann operator on smooth Jordan curves. Proc Am Math Soc 111(2):357–363, 1991).

Cite: Jollivet A. , Sharafutdinov V.
An Estimate for the Steklov Zeta Function of a Planar Domain Derived from a First Variation Formula
Journal of Geometric Analysis. 2022. V.32. N5. 161 . DOI: 10.1007/s12220-022-00890-7 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Feb 4, 2022
Accepted:	Feb 11, 2022
Published online:	Mar 9, 2022
Published print:	May 10, 2022

Identifiers:

Web of science:	WOS:000766588900001
Scopus:	2-s2.0-85126222047
Elibrary:	48190144
OpenAlex:	W3014425014

Citing:

DB	Citing
Scopus	2
Web of science	2
OpenAlex	2
Elibrary	1

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