{
  "id": "math/0606171",
  "version": "v1",
  "published": "2006-06-07T21:35:47.000Z",
  "updated": "2006-06-07T21:35:47.000Z",
  "title": "An inverse problem with data on the part of the boundary",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $u_t=\\nabla^2 u-q(x)u:=Lu$ in $D\\times [0,\\infty)$, where $D\\subset R^3$ is a bounded domain with a smooth connected boundary $S$, and $q(x)\\in L^2(S)$ is a real-valued function with compact support in $D$. Assume that $u(x,0)=0$, $u=0$ on $S_1\\subset S$, $u=a(s,t)$ on $S_2=S\\setminus S_1$, where $a(s,t)=0$ for $t>T$, $a(s,t)\\not\\equiv 0$, $a\\in C([0,T];H^{3/2}(S_2))$ is arbitrary. Given the extra data $u_N|_{S_2}=b(s,t)$, for each $a\\in C([0,T];H^{3/2}(S_2))$, where $N$ is the outer normal to $S$, one can find $q(x)$ uniquely. A similar result is obtained for the heat equation $u_t=\\mathcal{L} u:=%\\triangledown \\nabla \\cdot (a \\nabla u)$. These results are based on new versions of Property C.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-07T21:35:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K20",
      "35R30"
    ],
    "keywords": [
      "inverse problem",
      "heat equation",
      "smooth connected boundary",
      "extra data",
      "outer normal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6171R"
    }
  }
}