{
  "id": "1107.2306",
  "version": "v1",
  "published": "2011-07-12T14:46:27.000Z",
  "updated": "2011-07-12T14:46:27.000Z",
  "title": "Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian",
  "authors": [
    "Eleonora Cinti"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation $(-\\Delta)^{1/2}u=f(u)$ in all the space $\\re^{2m}$, where $f$ is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension $2m$, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4$ and $2m=6$. These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-12T14:46:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "35J20"
    ],
    "keywords": [
      "saddle-shaped solution",
      "bistable elliptic equations",
      "half-laplacian",
      "elliptic fractional equation",
      "high dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2306C"
    }
  }
}