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An estimate for the Steklov zeta function of a planar domain derived from a first variation formula.
| Content Provider | Semantic Scholar |
|---|---|
| Author | Jollivet, Alexandre Sharafutdinov, Vladimir A. |
| Copyright Year | 2020 |
| Abstract | We consider the Steklov zeta function $\zeta$ $\Omega$ of a smooth bounded simply connected planar domain $\Omega$ $\subset$ R 2 of perimeter 2$\pi$. We provide a first variation formula for $\zeta$ $\Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $\in$ (--1, 0) $\cup$ (0, 1), the difference $\zeta$ $\Omega$ (s) -- 2$\zeta$ R (s) is non-negative and is equal to zero if and only if $\Omega$ is a round disk ($\zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $\zeta$ $\Omega$ (s) -- 2$\zeta$ R (s) $\ge$ 0 for s $\in$ (--$\infty$, --1] $\cup$ (1, $\infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $\zeta$ $\Omega$ (0) = 2$\zeta$ R (0) obtained by Edward and Wu [1991]. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/2004.01779v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
