{
  "id": "1506.00940",
  "version": "v1",
  "published": "2015-06-02T16:15:00.000Z",
  "updated": "2015-06-02T16:15:00.000Z",
  "title": "Bounded solutions to the Allen-Cahn equation with level sets of any compact topology",
  "authors": [
    "Alberto Enciso",
    "Daniel Peralta-Salas"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $\\Sigma$ of R^d, with $d\\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of $\\Sigma$. More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-02T16:15:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "allen-cahn equation",
      "compact topology",
      "bounded solutions",
      "zero level set",
      "bounded entire solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150600940E"
    }
  }
}