{
  "id": "1602.03712",
  "version": "v1",
  "published": "2016-02-11T12:58:07.000Z",
  "updated": "2016-02-11T12:58:07.000Z",
  "title": "Stabilization via Homogenization",
  "authors": [
    "Marcus Waurick"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, $\\partial_t^2 u_n-\\partial_x^2 u_n = \\partial_t f$ and $u_n-\\partial_x^2 u_n= f$ on the respective spatial domains $\\bigcup_{j\\in \\{1,\\ldots,n\\}} \\big(\\frac{j-1}{n},\\frac{2j-1}{2n}\\big)$ and $\\bigcup_{j\\in \\{1,\\ldots,n\\}} \\big(\\frac{2j-1}{n},\\frac{j}{n}\\big)$. We show that $(u_n)_n$ converges weakly to $u$, which solves the exponentially stable limit equation $\\partial_t^2 u +2\\partial_t u + u -\\partial_x^2 u = f+\\partial_t f$ on $[0,1]$. If the elliptic equation is replaced by a parabolic one, the limit equation is \\emph{not} exponentially stable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-11T12:58:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35M10",
      "35B35",
      "35B27"
    ],
    "keywords": [
      "stabilization",
      "homogenization",
      "elliptic equation",
      "short note",
      "elliptic type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203712W"
    }
  }
}