论文标题
通过边界测量确定黑洞
Determination of black holes by boundary measurements
论文作者
论文摘要
对于带有时间独立的Lorentzian Metric的波方程，在$ \ mathbb {r} \ timesω$中考虑一个初始的有限值问题，其中$ x_0 \ in \ mathbb {r} $是time variable，$ x $是$ \ mathbbbb in $ \ m m i \ mathbb {r}^r}^n $。令$γ\ subset \partialΩ$为$ \partialΩ$的子域。我们说，如果我们知道$ \ Mathbb {r} \ timesγ$的dirichlet和neumann数据，则在$ \ mathbb {r} \timesγ$上给出了边界测量。逆边界值问题包括从边界数据中恢复度量。在作者的先前作品中，开发了边界控制方法的局部变体，该变体允许在$ω$的任何点的附近恢复局部度量，其中波算子的空间部分是椭圆形的。这允许恢复甲孔外部的度量。我们的目标是恢复黑洞。在某些情况下，崇高与黑洞相吻合。在两个空间尺寸的情况下，我们恢复了甲壳虫内部内部的黑洞，假设Ergosphere（即，在Ergosphere的任何点都不是特征）。
For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in $\mathbb{R}\times Ω$, where $x_0\in \mathbb{R}$, is the time variable and $Ω$ is a bounded domain in $\mathbb{R}^n$. Let $Γ\subset\partialΩ$ be a subdomain of $\partialΩ$. We say that the boundary measurements are given on $\mathbb{R}\timesΓ$ if we know the Dirichlet and Neumann data on $\mathbb{R}\times Γ$. The inverse boundary value problem consists of recovery of the metric from the boundary data. In author's previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of $Ω$ where the spatial part of the wave operator is elliptic. This allow the recovery of the metric in the exterior of the ergoregion. Our goal is to recover the black hole. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.
