TY - JOUR

T1 - An inequality for the Steklov spectral zeta function of a planar domain

AU - Jollivet, Alexandre

AU - Sharafutdinov, Vladimir

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Dirichlet-to-Neumann operator

KW - Inverse spectral problem

KW - Steklov spectrum

KW - Zeta function

KW - inverse spectral problem

KW - EIGENVALUES

KW - NEUMANN OPERATOR

KW - zeta function

UR - http://www.scopus.com/inward/record.url?scp=85042864545&partnerID=8YFLogxK

U2 - 10.4171/JST/196

DO - 10.4171/JST/196

M3 - Article

AN - SCOPUS:85042864545

VL - 8

SP - 271

EP - 296

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

SN - 1664-039X

IS - 1

ER -