{
  "id": "1001.4940",
  "version": "v1",
  "published": "2010-01-27T13:34:36.000Z",
  "updated": "2010-01-27T13:34:36.000Z",
  "title": "Inverse problem for wave equation with sources and observations on disjoint sets",
  "authors": [
    "Matti Lassas",
    "Lauri Oksanen"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "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 $1\\leq j<k\\leq 3$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-27T13:34:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30"
    ],
    "keywords": [
      "inverse problem",
      "disjoint sets",
      "wave equation",
      "restricted dirichlet-to-neumann operator",
      "riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010InvPr..26h5012L"
    }
  }
}