论文标题
在裂纹逆问题上，压力波在半空间中
On the crack inverse problem for pressure waves in half-space
论文作者
论文摘要
在半空间中制定压力波方程后，在顶面上的neumann条件为零的裂纹，我们引入了相关的逆问题。该反问题包括在相对开放的顶部平面集中识别该裂缝上的裂纹和未知的强迫项。除非做出一些额外的假设，否则这个逆问题无法唯一解决。但是，我们表明我们可以在$ \ rr^3 \ setMinus \ edromline {γ_1\ cupγ_2} $的假设下区分两个裂纹$γ_1$和$γ_2$。如果不是这种情况，我们提供的反例表现出了非唯一性，即使$γ_1$和$γ_2$平滑且“几乎”平坦。最后，我们显示在$ \ rr^3 \ setMinus \ edroline {γ_1\cupγ_2} $不一定连接的情况下，在排除了离散的频率之后，$γ_1$和$γ_2$可以再次与过度确定的边界数据区分开。
After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks $Γ_1$ and $Γ_2$ under the assumption that $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if $Γ_1$ and $Γ_2$ are smooth and "almost" flat. Finally, we show in the case where $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ is not necessarily connected that after excluding a discrete set of frequencies, $Γ_1$ and $Γ_2$ can again be differentiated from overdetermined boundary data.
