{
  "id": "1503.01103",
  "version": "v1",
  "published": "2015-03-03T20:59:38.000Z",
  "updated": "2015-03-03T20:59:38.000Z",
  "title": "On uniform estimates for Laplace equation in balls with small holes",
  "authors": [
    "Yong Lu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the Dirichlet problem of the Laplace equation in the unit ball with a shrinking hole. The problem typically arises from homogenization problems in domains perforated with very small holes. In three dimensions, we show that there holds the uniform $W^{1,p}$ estimate when $3/2<p<3$. We also show that the numbers $3/2$ and $3$ are critical in the sense that for any $1<p<3/2$ or $3<p<\\infty $, there are counterexamples indicating that the uniform $W^{1,p}$ estimate does not hold. The results can be generalized to higher dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-03T20:59:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplace equation",
      "small holes",
      "uniform estimates",
      "unit ball",
      "dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150301103L"
    }
  }
}