论文标题
在平面平面域的Steklov Zeta函数之间的差异范围与$2π$周围和单位磁盘之间的差异性能
Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2π$ perimeter and of the unit disk
论文作者
论文摘要
我们认为，对于简单连接的平面域$ω$的ZETA函数$ζ_Ω$，由光滑的封闭曲线限制为$2π$。我们证明$ζ_Ω''（0）\geζ_ {\ mathbb {d}}''（0）$ at equality at equality，并且仅当$ω$是$ \ mathbb {d} $的磁盘时，则表示关闭的单位磁盘。我们还提供了一个基本证据，表明满足$ s \ le-1 $的固定$ s $估计$ζ_Ω''（s）\geζ_ {\ mathbb {d}}}'（s）''（s）$在$ω$的情况下保持平等。然后，我们将域$ω$的示例带到了单位磁盘附近，该估计未能扩展到间隔$（0,2）$。文本的其余部分也详细介绍了与先前作品有关的其他计算。
We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ 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 $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ 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.
