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

TL;DR

This paper investigates the convexity properties of the difference between the Steklov zeta functions of a smooth planar domain and the unit disk, establishing inequalities and conditions for equality, with implications for spectral geometry.

Contribution

It proves new inequalities for the Steklov zeta function differences, characterizes when equality holds, and provides elementary proofs and examples related to spectral properties of planar domains.

Findings

Proved that ta_\u2207''(0)  ta_{\u25cb}''(0) with equality iff ta_\u2207 is a disk.

Established that for s  -1, ta_''(s)  ta_{\u25cb}''(s) with equality iff ta_ is a disk.

Provided examples of domains close to the disk where the inequalities fail to extend to (0,2).

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 $\zeta_\Omega''(0)\ge \zeta_{\mathbb{D}}''(0)$ with equality if and only if $\Omega$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $\zeta_\Omega''(s)\ge \zeta_{\mathbb{D}}''(s)$ 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.