{
  "id": "1603.06088",
  "version": "v1",
  "published": "2016-03-19T12:26:21.000Z",
  "updated": "2016-03-19T12:26:21.000Z",
  "title": "Fractional perimeter from a fractal perspective",
  "authors": [
    "Luca Lombardini"
  ],
  "comment": "4 figures. arXiv admin note: substantial text overlap with arXiv:1508.06241",
  "categories": [
    "math.AP"
  ],
  "abstract": "Following \\cite{Visintin}, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we give some examples of this kind of sets, explaining how to construct them starting from well known self-similar fractals. In particular, we show that in the case of the von Koch snowflake $S\\subset\\mathbb R^2$ this fractal dimension coincides with the Minkowski dimension, namely \\begin{equation*} P_s(S)<\\infty\\qquad\\Longleftrightarrow\\qquad s\\in\\Big(0,2-\\frac{\\log4}{\\log3}\\Big). \\end{equation*} We also study the asymptotics as $s\\to1^-$ of the fractional perimeter of a set having finite (classical) perimeter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-19T12:26:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional perimeter",
      "fractal perspective",
      "measure theoretic boundary",
      "von koch snowflake",
      "fractal dimension coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}