Convexity properties of the normalized Steklov zeta function of a planar domain

Alexandre Jollivet (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

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 of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge \zeta_{\D}''(0)$ with equality if and only if $\Omega$ is a disk where $\D$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $\zeta_\Omega''(s)\ge \zeta_{\D}''(s)$ holds with equality if and only if $\Omega$ is a disk. We then bring examples of domains $\Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Alexandre Jollivet. Convexity properties of the normalized Steklov zeta function of a planar domain. Journal of Inverse and Ill-posed Problems, 2021, ⟨10.1515/jiip-2020-0113⟩. ⟨hal-02918987⟩