{
  "id": "math/0504320",
  "version": "v2",
  "published": "2005-04-15T12:48:56.000Z",
  "updated": "2005-05-25T11:35:06.000Z",
  "title": "Maxwell's Equations with Scalar Impedance: Inverse Problems with data given on a part of the boundary",
  "authors": [
    "Yaroslav Kurylev",
    "Matti Lassas",
    "Erkki Somersalo"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study Maxwell's equations in time domain in an anisotropic medium. The goal of the paper is to solve an inverse boundary value problem for anisotropies characterized by scalar impedance $\\alpha$. This means that the material is conformal, i.e., the electric permittivity $\\epsilon$ and magnetic permeability $\\mu$ are tensors satisfying $\\mu =\\alpha^2\\epsilon$. This condition is equivalent to a single propagation speed of waves with different polarizations which uniquely defines an underlying Riemannian structure. The analysis is based on an invariant formulation of the system of electrodynamics as a Dirac type first order system on a Riemannian $3-$manifold with an additional structure of the wave impedance, $(M,g,\\alpha)$, where $g$ is the travel-time metric. We study the properties of this system in the first part of the paper. In the second part we consider the inverse problem, that is, the determination of $(M,g,\\alpha)$ from measurements done only on an open part of the boundary and on a finite time interval. As an application, in the isotropic case with $M\\subset \\R^3$, we prove that the boundary data given only on an open part of the boundary determine uniquely the domain $M$ and the coefficients $\\epsilon$ and $\\mu$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-05-25T11:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J45",
      "35R30",
      "35Q60"
    ],
    "keywords": [
      "scalar impedance",
      "maxwells equations",
      "inverse problem",
      "dirac type first order system",
      "inverse boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4320K"
    }
  }
}