Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2\pi$ perimeter and of the unit disk - Archive ouverte HAL Access content directly
Journal Articles Journal of Inverse and Ill-posed Problems Year : 2021

Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2\pi$ perimeter and of the unit disk

Alexandre Jollivet
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Abstract

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.
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Dates and versions

hal-02918987 , version 1 (21-08-2020)

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Alexandre Jollivet. Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2\pi$ perimeter and of the unit disk. Journal of Inverse and Ill-posed Problems, 2021, ⟨10.1515/jiip-2020-0113⟩. ⟨hal-02918987⟩
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