{
  "id": "1012.2741",
  "version": "v1",
  "published": "2010-12-13T14:37:24.000Z",
  "updated": "2010-12-13T14:37:24.000Z",
  "title": "Fractional Harmonic Maps into Manifolds in odd dimension n>1",
  "authors": [
    "Francesca Da Lio"
  ],
  "comment": "27 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider critical points of the following nonlocal energy {equation} {\\cal{L}}_n(u)=\\int_{\\R^n}| ({-\\Delta})^{n/4} u(x)|^2 dx\\,, {equation} where $u\\colon H^{n/2}(\\R^n)\\to{\\cal{N}}\\,$ ${\\cal{N}}\\subset\\R^m$ is a compact $k$ dimensional smooth manifold without boundary and $n>1$ is an odd integer. Such critical points are called $n/2$-harmonic maps into ${\\cal{N}}$. We prove that $\\Delta ^{n/2} u\\in L^p_{loc}(\\R^n)$ for every $p\\ge 1$ and thus $u\\in C^{0,\\alpha}_{loc}(\\R^n)\\,.$ The local H\\\"older continuity of $n/2$-harmonic maps is based on regularity results obtained in \\cite{DL1} for nonlocal Schr\\\"odinger systems with an antisymmetric potential and on suitable {\\it 3-terms commutators} estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-13T14:37:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E20",
      "35J20",
      "35B65",
      "35J60",
      "35S99"
    ],
    "keywords": [
      "fractional harmonic maps",
      "odd dimension",
      "critical points",
      "dimensional smooth manifold",
      "nonlocal energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.2741D"
    }
  }
}