论文标题
从第一个变化公式得出的平面域的steklov zeta函数的估计值
An estimate for the Steklov zeta function of a planar domain derived from a first variation formula
论文作者
论文摘要
我们认为平滑界的简单连接的平面域$ω$ $ $ \ subset $ r 2的steklov zeta函数$ζ$ $ $ω$ 2 $π$。在域的平滑变形下，我们为$ζ$ $ω$提供了第一个变化公式。在公式的基础上，我们证明，对于$（-1，0）$ \ cup $（0，1）中的每个s $ \，差异$ζ$ $ $ $ $ $ $ ch $（s）-2 $ζ$ r（s）是非阴性的，并且当$ω$是一个回合disk（$ q q qum qζ$ ri是ZET的Zetal Riemann zeta riiemann finfactial zeran rymann zere and y等于零）。 Our approach gives also an alternative proof of the inequality $ζ$ $Ω$ (s) -- 2$ζ$ 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 $ζ$ $Ω$ (0) = 2 $ζ$ r（0）由Edward and Wu [1991]获得。
We consider the Steklov zeta function $ζ$ $Ω$ of a smooth bounded simply connected planar domain $Ω$ $\subset$ R 2 of perimeter 2$π$. We provide a first variation formula for $ζ$ $Ω$ 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 $ζ$ $Ω$ (s) -- 2$ζ$ R (s) is non-negative and is equal to zero if and only if $Ω$ is a round disk ($ζ$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $ζ$ $Ω$ (s) -- 2$ζ$ 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 $ζ$ $Ω$ (0) = 2$ζ$ R (0) obtained by Edward and Wu [1991].
