Inverse problem for wave equation with sources and observations on disjoint sets

Abstract

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that ${\Gamma }_1$ and ${\Gamma }_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator ${\Lambda }_{{\Gamma }_1,{\Gamma }_2}$. This operator corresponds the boundary measurements when we have smooth sources supported on ${\Gamma }_1$ and the fields produced by these sources are observed on ${\Gamma }_2$. We show that when ${\Gamma }_1$ and ${\Gamma }_2$ are disjoint but their closures intersect at least at one point, then the restricted Dirichlet-to-Neumann operator ${\Lambda }_{{\Gamma }_1,{\Gamma }_2}$ 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 ${\Gamma }_1,{\Gamma }_2$, and ${\Gamma }_3$ of the boundary, then the restricted Dirichlet-to-Neumann operators ${\Lambda }_{{\Gamma }_j,{\Gamma }_k}$ for $1\le j<k\le 3$ 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.

References in corpus (1)

• Limiting Carleman weights and anisotropic inverse problems

Cited by in corpus (5)

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• Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds