An inequality for the Steklov spectral zeta function of a planar domain

An inequality for the Steklov spectral zeta function of a planar domain Full article

Journal

Journal of Spectral Theory
ISSN: 1664-039X , E-ISSN: 1664-0403

Output data

Year: 2018, Volume: 8, Number: 1, Pages: 271-296 Pages count : 26 DOI: 10.4171/JST/196

Tags

Dirichlet-to-Neumann operator; Inverse spectral problem; Steklov spectrum; Zeta function

Authors

Jollivet A. ¹ , Sharafutdinov V. ^2,3

Affiliations

1	Laboratoire de Mathématiques Paul Painlevé, CNRS UMR 8524, Université Lille 1 Sciences et Technologies, Villeneuve d'Ascq Cedex, 59655, France
2	Department of Mathematics and Mechanics, Novosibirsk State University, Pirogova St. 2, Novosibirsk, 630090, Russian Federation
3	Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, 630090, Russian Federation

We consider the zeta function Ω for the Dirichlet-to-Neumann operator of a simply connected planar domain Ωbounded by a smooth closed curve. We prove that, for a fixed real s satisfying jsj > 1 and fixed length L.@ Ω/ of the boundary curve, the zeta function Ω.s/ reaches its unique minimum when Ωis a disk. This result is obtained by studying the difference Ω(s)-2L.@ Ω/ 2 π R.s/,where R stands for the classicalRiemann zeta function. The difference turns out to be non-negative for real s satisfying jsj > 1. We prove some growth properties of the difference as s →±∞ Two analogs of these results are also provided.

Jollivet A. , Sharafutdinov V.
An inequality for the Steklov spectral zeta function of a planar domain
Journal of Spectral Theory. 2018. V.8. N1. P.271-296. DOI: 10.4171/JST/196 WOS Scopus OpenAlex

Web of science:	WOS:000424290700007
Scopus:	2-s2.0-85042864545
OpenAlex:	W2793000411

DB	Citing
Scopus	5
OpenAlex	6
Web of science	5