不相交集上信源和观测值为wave equation的反问题
Inverse problem for wave equation with sources and observations on disjoint sets
摘要：
考虑紧致黎曼流形上双曲型偏微分方程的反问题。假设$Γ_1$和$Γ_2$是流形边界的两个不相交的开子集，我们定义了限制Dirichlet-to-Neumann算子$Λ_{Γ_1,Γ_2}$。当我们在$Γ_1$上支持平滑源并且在$Γ_2$上观察这些源产生的场时，该算子对应于边界测量。我们证明了当$Γ_1$和$Γ_2$是不相交的，但他们的闭包相交至少在一个点，然后限制Dirichlet到诺依曼算子$Λ_{Γ_1,Γ_2}$确定黎曼流形和度量它的等距。在欧几里德空间中，结果得到，即使波源远离接收器，通过在物体的边界上的测量，也可以确定致密物体内部的各向异性波速，直到自然的坐标变换。此外，我们证明了如果我们有三个任意的非空开子集$Γ_1,Γ_2$，和$Γ_3$的边界，那么限制Dirichlet-to-Neumann算子$Λ_{Γ_j,Γ_k}$的$1\leq j<k\leq 3$确定黎曼流形的等距.当双曲型方程满足精确能控性条件时，对于有限时间边界测量也证明了类似的结果。
英文摘要：
We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $Γ_1$ and $Γ_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator $Λ_{Γ_1,Γ_2}$. This operator corresponds the boundary measurements when we have smooth sources supported on $Γ_1$ and the fields produced by these sources are observed on $Γ_2$. We show that when $Γ_1$ and $Γ_2$ are disjoint but their closures intersect at least at one point, then the restricted Dirichlet-to-Neumann operator $Λ_{Γ_1,Γ_2}$ determines the Riemannian manifold and the metric on it up to an isometry. In the Euclidian space, the result yields that an anisotropic wave speed inside a compact body is determined, up to a natural coordinate transformations, by measurements on the boundary of the body even when wave sources are kept away from receivers. Moreover, we show that if we have three arbitrary non-empty open subsets $Γ_1,Γ_2$, and $Γ_3$ of the boundary, then the restricted Dirichlet-to-Neumann operators $Λ_{Γ_j,Γ_k}$ for $1\leq j<k\leq 3$ determine the Riemannian manifold to an isometry. Similar result is proven also for the finite-time boundary measurements when the hyperbolic equation satisfies an exact controllability condition.
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