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

Alexandre Jollivet (1) , Vladimir Sharafutdinov (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Sobolev Institute of Mathematics

We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve. We prove non-negativeness and growth properties for $\zeta_\Omega(s)-2\big({L(\partial \Omega)\over 2\pi}\big)^s\zeta_R(s)\ (s\leq-1)$, where $L(\partial \Omega)$ is the length of the boundary curve and $\zeta_R$ stands for the classical Riemann zeta function. Two analogs of these results are also provided.

Mots clés: Dirichlet-to-Neumann operator; zeta function; inverse spectral problem; Steklov spectrum

Domaines: Physique mathématique [math-ph]

Alexandre Jollivet, Vladimir Sharafutdinov. An inequality for the Steklov spectral zeta function of a planar domain. Journal of Spectral Theory, 2018, 8 (1), pp.271-296. ⟨10.4171/jst/196⟩. ⟨hal-01219201⟩