{
  "id": "1308.6814",
  "version": "v1",
  "published": "2013-08-30T18:14:35.000Z",
  "updated": "2013-08-30T18:14:35.000Z",
  "title": "Extensions and their Minimizations on the Sierpinski Gasket",
  "authors": [
    "Pak Hin Li",
    "Nicholas Ryder",
    "Robert S. Strichartz",
    "Baris Evren Ugurcan"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional $\\mathcal{E}_{\\lambda}(f) = \\mathcal{E}(f,f) + \\lambda \\int f^2 d \\mu$ with prescribed values at a finite set of points where $\\mathcal{E}$ denotes the energy (the analog of $\\int |\\nabla f|^2$ in Euclidean space) and $\\mu$ denotes the standard self-similiar measure on $SG$. We explicitly construct the minimizer $f(x) = \\sum_{i} c_i G_{\\lambda}(x_i, x)$ for some constants $c_i$, where $G_{\\lambda}$ is the resolvent for $\\lambda \\geq 0$. We minimize the energy over sets in $SG$ by calculating the explicit quadratic form $\\mathcal{E}(f)$ of the minimizer $f$. We consider properties of this quadratic form for arbitrary sets and then analyze some specific sets. One such set we consider is the bottom row of a graph approximation of $SG$. We describe both the quadratic form and a discretized form in terms of Haar functions which corresponds to the continuous result established in a previous paper. In the second part, we study a similar problem this time minimizing $\\int_{SG} |\\Delta f(x)|^2 d \\mu (x)$ for general measures. In both cases, by using standard methods we show the existence and uniqueness to the minimization problem. We then study properties of the unique minimizers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-30T18:14:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80"
    ],
    "keywords": [
      "sierpinski gasket",
      "minimization",
      "standard self-similiar measure",
      "explicit quadratic form",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.6814H"
    }
  }
}