{
  "id": "1411.1697",
  "version": "v1",
  "published": "2014-11-06T18:41:59.000Z",
  "updated": "2014-11-06T18:41:59.000Z",
  "title": "A direct method of moving planes for the fractional Laplacian",
  "authors": [
    "Wenxiong Chen",
    "Congming Li",
    "Yan Li"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we develop a direct method of moving planes for the fractional Laplacian. Instead of conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using a simple example, a semi-linear equation involving the fractional Laplacian, we illustrate how this new method of moving planes can be employed to obtain symmetry and non-existence of positive solutions. We firmly believe that the ideas introduced here can be applied to more general non-local operators with gener",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-06T18:41:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60"
    ],
    "keywords": [
      "fractional laplacian",
      "moving planes",
      "direct method",
      "general non-local operators",
      "conventional extension method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.1697C"
    }
  }
}