{
  "id": "1509.02693",
  "version": "v1",
  "published": "2015-09-09T09:38:58.000Z",
  "updated": "2015-09-09T09:38:58.000Z",
  "title": "Conformal mapping for cavity inverse problem: an explicit reconstruction formula",
  "authors": [
    "Alexandre Munnier",
    "Karim Ramdani"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we address a classical case of the Calder\\'on (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity $\\omega$ (with boundary $\\gamma$) contained in a domain $\\Omega$ (with boundary $\\Gamma$) from the knowledge of the Dirichlet-to-Neumann (DtN) map $\\Lambda_\\gamma: f \\longmapsto \\partial_n u^f|_{\\Gamma}$, where $u^f$ is harmonic in $\\Omega\\setminus\\overline{\\omega}$, $u^f|_{\\Gamma}=f$ and $u^f|_{\\gamma}=c^f$, $c^f$ being the constant such that $\\int_{\\gamma}\\partial_n u^f\\,{\\rm d}s=0$. We obtain an explicit formula for the complex coefficients $a_m$ arising in the expression of the Riemann map $z\\longmapsto a_1 z + a_0 + \\sum_{m\\leqslant -1} a_m z^{m}$ that conformally maps the exterior of the unit disk onto the exterior of $\\omega$. This formula is derived by using two ingredients: a new factorization result of the DtN map and the so-called generalized P\\'olia-Szeg\\\"o tensors (GPST) of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients $a_m$ with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-09T09:38:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit reconstruction formula",
      "cavity inverse problem",
      "conformal mapping",
      "analytic dependence",
      "single cavity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}